I am getting closer - now at least there are no error messages.  Unfortunately, now my applet only draws a light-grey square.  I assume this has something to do with my not using the Canvas correctly.  Here is my code, in case you can see something I've obviously done incorrectly:

import java.awt.BorderLayout;
import java.awt.Canvas;
import java.awt.Dimension;
import java.awt.Graphics;
import java.awt.event.MouseEvent;
import java.awt.event.MouseListener;
import java.awt.event.MouseMotionListener;

import java.io.InputStream;

import javax.swing.JApplet;

import com.threed.jpct.Camera;
import com.threed.jpct.FrameBuffer;
import com.threed.jpct.IRenderer;
import com.threed.jpct.Lights;
import com.threed.jpct.Loader;
import com.threed.jpct.Matrix;
import com.threed.jpct.Object3D;
import com.threed.jpct.SimpleVector;
import com.threed.jpct.World;

import java.lang.System;

public class jPCTGears extends JApplet implements MouseListener, MouseMotionListener
{
    private Object3D redGear, greenGear, blueGear, assemblyPivot;
    private FrameBuffer buffer = null;
    private World world = null;
    private Camera camera = null;
    private int width = 640;
    private int height = 480;
    private float gear_rotation = 0.02f;
    private int prevMouseX, prevMouseY;
   
    Canvas myCanvas;
   
    // Initialize all components of the applet
    @Override
    public void init()
    {
        // sign the applet up to receive mouse messages:
        addMouseListener( this );
        addMouseMotionListener( this );
       
        world = new World();  // create a new world
       
        World.setDefaultThread( Thread.currentThread() );

        // create a new buffer to draw on:
        buffer = new FrameBuffer( width, height, FrameBuffer.SAMPLINGMODE_HARDWARE_ONLY );
        buffer.enableRenderer( IRenderer.RENDERER_OPENGL );
        buffer.disableRenderer( IRenderer.RENDERER_SOFTWARE );
       
        myCanvas = buffer.enableGLCanvasRenderer();
        add( myCanvas, BorderLayout.CENTER);
        myCanvas.setVisible( true );
       
        // load some 3D objects and make sure they have the correct orientation:
        redGear = loadMeshFile( "RedGear.3ds" );
        redGear.rotateY( (float)Math.PI / 2.0f );
        redGear.rotateMesh();
        redGear.setRotationMatrix( new Matrix() );
        redGear.setOrigin( new SimpleVector( 0, 0, 0 ) );
        redGear.build();
        greenGear = loadMeshFile( "GreenGear.3ds" );
        greenGear.rotateY( (float)Math.PI / 2.0f );
        greenGear.rotateZ( 0.35f );
        greenGear.rotateMesh();
        greenGear.setRotationMatrix( new Matrix() );
        greenGear.setOrigin( new SimpleVector( -145.0f, 0, 0 ) );
        greenGear.build();
        blueGear = loadMeshFile( "BlueGear.3ds" );
        blueGear.rotateY( (float)Math.PI / 2.0f );
        //blueGear.rotateZ( 0.40f );
        blueGear.rotateMesh();
        blueGear.setRotationMatrix( new Matrix() );
        blueGear.setOrigin( new SimpleVector( 0, -140.0f, 0 ) );
        blueGear.build();

        // Set up a pivot point for the entire gear assembly:
        assemblyPivot = Object3D.createDummyObj();
        assemblyPivot.setOrigin( new SimpleVector( 0, 0, 0 ) );
        // Make the gears children to assemblyPivot.
        // Translations and rotations to assemblyPivot
        // will affect the entire gear assembly:
        assemblyPivot.addChild( redGear );
        assemblyPivot.addChild( greenGear );
        assemblyPivot.addChild( blueGear );
               
        // add the objects our world:
        world.addObject( redGear );
        world.addObject( greenGear );
        world.addObject( blueGear );
        world.buildAllObjects();
       
        lookAt( redGear );  // make sure the camera is facing towards the object
        letThereBeLight();  // create light sources for the scene
    }

    // Draw the scene
    @Override
    public void paint( Graphics g )
    {
        // rotate the gears:
        redGear.rotateAxis( redGear.getZAxis(), -gear_rotation );
        greenGear.rotateAxis( greenGear.getZAxis(), 2.0f * gear_rotation );
        blueGear.rotateAxis( blueGear.getZAxis(), 2.0f * gear_rotation );
       
        buffer.clear();   // erase the previous frame

        // render the world onto the buffer:
        world.renderScene( buffer );
        world.draw( buffer );
        buffer.update();
       
        buffer.displayGLOnly();
        myCanvas.repaint( 200, 0, 0, width, height );
    }

    // Load a 3Ds file, and return its Object3D handle
    public Object3D loadMeshFile( String filename )
    {
        Object3D newObject;
        //Object3D[] objs = Loader.load3DS( myURL, "models" + "/" + filename, 1.0f );
        Object3D[] objs = Loader.load3DS( getResource( filename ), 1.0f );

        if( objs.length==1 )
        {
            // The object loaded fine, just need to initialize it
            newObject=objs[0];
            newObject.setCulling( Object3D.CULLING_DISABLED );
            newObject.build();
        }
        else
        {
            // Didn't load anything, or loaded
            //     more than 1 object (not supposed to happen)
            System.out.println( "Unknown file format: " + filename );
            newObject = null;
        }

        return newObject;
    }
   
    private InputStream getResource( String resourceName )
    {
        return getClass().getClassLoader().getResourceAsStream( resourceName );
    }
    // point the camera toward the given object
    private void lookAt( Object3D obj )
    {
        camera = world.getCamera();  // grab a handle to the camera
        camera.setPosition( 0, 0, 500 );  // set its *relative* position
        camera.lookAt( obj.getTransformedCenter() );  // look toward the object
    }

    // create light sources for the scene
    private void letThereBeLight()
    {
        world.getLights().setOverbrightLighting (
            Lights.OVERBRIGHT_LIGHTING_DISABLED );
        world.getLights().setRGBScale( Lights.RGB_SCALE_2X );

        // Set the overall brightness of the world:
        world.setAmbientLight( 50, 50, 50 );

        // Create a main light-source:
        world.addLight( new SimpleVector( 50, -50, 300 ), 20, 20, 20 );
    }
   
    // Dragging the mouse should rotate the entire gear assembly
    public void mouseDragged( MouseEvent e )
    {
        // get the mouse's coordinates:
        int x = e.getX();
        int y = e.getY();
        Dimension size = e.getComponent().getSize();

        // Calculate the angles to rotate the assembly:
        float thetaY = (float)Math.PI * ( (float)(x-prevMouseX)/(float)size.width );
        float thetaX = (float)Math.PI * ( (float)(prevMouseY-y)/(float)size.height );
       
        // Apply the rotations to the gear assembly:
        assemblyPivot.rotateAxis( assemblyPivot.getXAxis(), thetaX );
        assemblyPivot.rotateAxis( assemblyPivot.getYAxis(), thetaY );

        // Keep track of the mouse location:
        prevMouseX = x;
        prevMouseY = y;
    }
    public void mousePressed( MouseEvent e )
    {
        // Start keeping track of the mouse location now.
        // This prevent the gear assembly from jerking to the new angle
        // whenever mouseDragged first gets called:
        prevMouseX = e.getX();
        prevMouseY = e.getY();
    }
    public void mouseReleased( MouseEvent e ){}
    public void mouseEntered( MouseEvent e ) {}
    public void mouseExited( MouseEvent e ) {}
    public void mouseClicked( MouseEvent e ) {}
    public void mouseMoved( MouseEvent e ) {}
}

BTW, I appreciate the assistance you've been giving me.

I am having a little trouble getting my applet to run in SAMPLINGMODE_HARDWARE_ONLY.
The Sun Java Console shows:
Java version is: 1.6.0_03
-> support for BufferedImage
Version helper for 1.2+ initialized!
-> using BufferedImage
Software renderer (OpenGL mode) initialized
java.lang.NoClassDefFoundError: org/lwjgl/opengl/Display
at com.threed.jpct.GLHelper.findMode(Unknown Source)
at com.threed.jpct.GLHelper.init(Unknown Source)
at com.threed.jpct.GLRenderer.init(Unknown Source)
at com.threed.jpct.FrameBuffer.enableRenderer(Unknown Source)
at com.threed.jpct.FrameBuffer.enableRenderer(Unknown Source)
at com.threed.jpct.FrameBuffer.enableRenderer(Unknown Source)
at jPCTGears.init(jPCTGears.java:48)
at sun.applet.AppletPanel.run(Unknown Source)
at java.lang.Thread.run(Unknown Source)

Here is the code in question:
    // Initialize all components of the applet
    @Override
    public void init()
    {
        // sign the applet up to receive mouse messages:
        addMouseListener( this );
        addMouseMotionListener( this );
       
        world = new World();  // create a new world
       
        World.setDefaultThread(Thread.currentThread());

        // create a new buffer to draw on:
        buffer = new FrameBuffer( width, height, FrameBuffer.SAMPLINGMODE_HARDWARE_ONLY );
        buffer.enableRenderer( IRenderer.RENDERER_OPENGL );
        buffer.disableRenderer( IRenderer.RENDERER_SOFTWARE );

        // load some 3D objects and make sure they have the correct orientation:
        redGear = loadMeshFile( "RedGear.3ds" );
        redGear.rotateY( (float)Math.PI / 2.0f );
        redGear.rotateMesh();
        redGear.setRotationMatrix( new Matrix() );
        redGear.setOrigin( new SimpleVector( 0, 0, 0 ) );
        redGear.build();
        greenGear = loadMeshFile( "GreenGear.3ds" );
        greenGear.rotateY( (float)Math.PI / 2.0f );
        greenGear.rotateZ( 0.35f );
        greenGear.rotateMesh();
        greenGear.setRotationMatrix( new Matrix() );
        greenGear.setOrigin( new SimpleVector( -145.0f, 0, 0 ) );
        greenGear.build();
        blueGear = loadMeshFile( "BlueGear.3ds" );
        blueGear.rotateY( (float)Math.PI / 2.0f );
        blueGear.rotateZ( 0.40f );
        blueGear.rotateMesh();
        blueGear.setRotationMatrix( new Matrix() );
        blueGear.setOrigin( new SimpleVector( 0, 135.0f, 0 ) );
        blueGear.build();

        // Set up a pivot point for the entire gear assembly:
        //assemblyPivot = new Object3D(1);
        assemblyPivot = Object3D.createDummyObj();
        assemblyPivot.setOrigin( new SimpleVector( 0, 0, 0 ) );
        // Make the gears children to assemblyPivot.
        // Translations and rotations to assemblyPivot
        // will affect the entire gear assembly:
        assemblyPivot.addChild( redGear );
        assemblyPivot.addChild( greenGear );
        assemblyPivot.addChild( blueGear );
               
        // add the objects our world:
        world.addObject( redGear );
        world.addObject( greenGear );
        world.addObject( blueGear );
        world.buildAllObjects();
       
        lookAt( redGear );  // make sure the camera is facing towards the object
        letThereBeLight();  // create light sources for the scene
    }

The applet only works when I use SAMPLINGMODE_NORMAL and RENDERER_SOFTWARE.  I am sure it is something simple I am doing incorrectly.  I am just having a little trouble figuring out what.

Yep, that is definitely the trick.  I had actually solved the problem with a rotation + translation, but this way is a lot cleaner.

I did notice something odd, though.

When I use Object3D.createDummyObj() to make the dummy object, the world looks kind of funky, as though a number of polygons are not being drawn.

Using "new Object3D(0)" instead of "createDummyObj()" also causes the same behavior.

However, using "new Object3D(1)" works perfectly.  There is probably a perfectly reasonable explanation for this behavior, but I thought I would mention it since it did look a bit odd to me.

Here is my source code for reference:
import java.awt.Dimension;
import java.awt.Graphics;
import java.awt.event.MouseEvent;
import java.awt.event.MouseListener;
import java.awt.event.MouseMotionListener;

import java.io.File;

import javax.swing.JApplet;

import com.threed.jpct.Camera;
import com.threed.jpct.FrameBuffer;
import com.threed.jpct.IRenderer;
import com.threed.jpct.Lights;
import com.threed.jpct.Loader;
import com.threed.jpct.Matrix;
import com.threed.jpct.Object3D;
import com.threed.jpct.SimpleVector;
import com.threed.jpct.World;

import java.lang.System;

public class jPCTGears extends JApplet implements MouseListener, MouseMotionListener
{
    private Object3D redGear, blueGear, assemblyPivot;
    private FrameBuffer buffer = null;
    private World world = null;
    private Camera camera = null;
    private int width = 640;
    private int height = 480;
    private float gear_rotation = 0.02f;
    private int prevMouseX, prevMouseY;
   
    // Initialize all components of the applet
    @Override
    public void init()
    {
        // sign the applet up to receive mouse messages:
        addMouseListener( this );
        addMouseMotionListener( this );
       
        world = new World();  // create a new world

        // create a new buffer to render the world on:
        buffer = new FrameBuffer( width, height, FrameBuffer.SAMPLINGMODE_NORMAL );
        buffer.enableRenderer( IRenderer.RENDERER_SOFTWARE );               

        // load some 3D objects and make sure they have the correct orientation:
        redGear = loadMeshFile( "RedGear.3ds" );
        redGear.rotateY( (float)Math.PI / 2.0f );
        redGear.rotateMesh();
        redGear.setRotationMatrix( new Matrix() );
        redGear.setOrigin( new SimpleVector( 0, 0, 0 ) );
        redGear.build();
        blueGear = loadMeshFile( "BlueGear.3ds" );
        blueGear.rotateY( (float)Math.PI / 2.0f );
        blueGear.rotateZ( 0.35f );
        blueGear.rotateMesh();
        blueGear.setRotationMatrix( new Matrix() );
        blueGear.setOrigin( new SimpleVector( 0, 135.0f, 0 ) );
        blueGear.build();
       
        // Set up a pivot point for the entire gear assembly:
        assemblyPivot = new Object3D(1);
        assemblyPivot.setOrigin( new SimpleVector( 0, 0, 0 ) );
        // Make the gears children to assemblyPivot.
        // Translations and rotations to assemblyPivot
        // will affect the entire gear assembly:
        assemblyPivot.addChild( redGear );
        assemblyPivot.addChild( blueGear );
               
        // add the objects our world:
        world.addObject( redGear );
        world.addObject( blueGear );
        world.buildAllObjects();
       
        lookAt( redGear );  // make sure the camera is facing towards the gear assembly
        letThereBeLight();  // create light sources for the scene
    }

    // Draw the scene
    @Override
    public void paint( Graphics g )
    {
        // rotate the gears:
        redGear.rotateAxis( redGear.getZAxis(), -gear_rotation );
        blueGear.rotateAxis( blueGear.getZAxis(), 2.0f * gear_rotation );
       
        buffer.clear();   // erase the previous frame

        // render the world onto the buffer:
        world.renderScene( buffer );
        world.draw( buffer );
        buffer.update();

        buffer.display(g, 0, 0);  // draw the buffer onto the applet frame
       
        repaint( 200, 0, 0, width, height );  // keep the graphics auto-refreshing
    }

    // Load a 3Ds file, and return its Object3D handle
    private Object3D loadMeshFile( String filename )
    {
        Object3D newObject;
        Object3D[] objs = Loader.load3DS( this.getDocumentBase(), "models" + File.separatorChar + filename, 1.0f );

        if( objs.length==1 )
        {
            // The object loaded fine, just need to initialize it
            newObject=objs[0];
            newObject.setCulling( Object3D.CULLING_DISABLED );
            newObject.build();
        }
        else
        {
            // Didn't load anything, or loaded
            //     more than 1 object (not supposed to happen)
            System.out.println( "Unknown file format: " + filename );
            newObject = null;
        }

        return newObject;
    }

    // point the camera toward the given object
    private void lookAt( Object3D obj )
    {
        camera = world.getCamera();  // grab a handle to the camera
        camera.setPosition( 0, 0, 500 );  // set its *relative* position
        camera.lookAt( obj.getTransformedCenter() );  // look toward the object
    }

    // create light sources for the scene
    private void letThereBeLight()
    {
        world.getLights().setOverbrightLighting (
            Lights.OVERBRIGHT_LIGHTING_DISABLED );
        world.getLights().setRGBScale( Lights.RGB_SCALE_2X );

        // Set the overall brightness of the world:
        world.setAmbientLight( 50, 50, 50 );

        // Create a main light-source:
        world.addLight( new SimpleVector( 50, -50, 300 ), 20, 20, 20 );
    }
   
    // Dragging the mouse should rotate the entire gear assembly
    public void mouseDragged( MouseEvent e )
    {
        // get the mouse's coordinates:
        int x = e.getX();
        int y = e.getY();
        Dimension size = e.getComponent().getSize();

        // Calculate the angles to rotate the assembly:
        float thetaY = (float)Math.PI * ( (float)(x-prevMouseX)/(float)size.width );
        float thetaX = (float)Math.PI * ( (float)(prevMouseY-y)/(float)size.height );
       
        // Apply the rotations to the gear assembly:
        assemblyPivot.rotateAxis( assemblyPivot.getXAxis(), thetaX );
        assemblyPivot.rotateAxis( assemblyPivot.getYAxis(), thetaY );

        // Keep track of the mouse location:
        prevMouseX = x;
        prevMouseY = y;
    }
    public void mousePressed( MouseEvent e )
    {
        // Start keeping track of the mouse location now.
        // This prevent the gear assembly from jerking to the new angle
        // whenever mouseDragged first gets called:
        prevMouseX = e.getX();
        prevMouseY = e.getY();
    }
    public void mouseReleased( MouseEvent e ){}
    public void mouseEntered( MouseEvent e ) {}
    public void mouseExited( MouseEvent e ) {}
    public void mouseClicked( MouseEvent e ) {}
    public void mouseMoved( MouseEvent e ) {}
}

At any rate, my question is solved, and I learned a little bit along the way.  Time for me to try something more difficult    Thanks a lot!

Each frame, I am trying to change the rotation pivot of an object from (0,-135,0) to (0,0,0) before doing a rotateAxis() on it, and then change the rotation pivot back to (0,-135,0) afterwards.  It seems like the following code should work:

        myObject.setRotationPivot( new SimpleVector( 0, 0, 0 ) );
        myObject.rotateAxis( myObject.getZAxis(), radZ );
        myObject.setRotationPivot( new SimpleVector( 0, -135.0f, 0 ) );

Instead, the object just rotates around the original pivot (0,-135,0), the same as if I don't put any setRotationPivot()'s in there at all.  I am sure there is a simple solution to this, but I can't seem figure it out.