Sounds like you need picking. It's part of the api. In 3d a ray is projected
into the space and you will get back all elements that intersected that
(invisible) ray (e.g. your reversi cells if they are instances of Shape3D).

an example from an old project and poorly documented:

public class FPPicker extends Behavior
{
   public final String ACTION_COMMAND = "VertexPicked";
   
   protected PickCanvas pickCanvas;

   protected WakeupOnAWTEvent wakeupCriterion;

   protected int x;
   protected int y;

   protected FPPickListener fpp;
   
   /** Creates a new instance of FPPicker */
   public FPPicker(Canvas3D canvas, BranchGroup scene)
   {
       //FPPListenerList = new ArrayList();
       pickCanvas = new PickCanvas(canvas, scene);
       // Objects within this disstance from the mouse location will be
picked.
       pickCanvas.setTolerance(3.0f);
       // We want to get info (indices of points) from the intersected
geometry (the PointArray).
       pickCanvas.setMode(PickCanvas.BOUNDS);
   }
   
   public void initialize()
   {
       // Wake up when a mouse button is clicked
       wakeupCriterion = new WakeupOnAWTEvent(MouseEvent.MOUSE_CLICKED);
       wakeupOn(wakeupCriterion);
   }

   public void processStimulus(Enumeration criteria)
   {
       while (criteria.hasMoreElements())
       {
           WakeupCriterion wakeup =
(WakeupCriterion)criteria.nextElement();
           if (wakeup instanceof WakeupOnAWTEvent)
           {
               AWTEvent[] events =
((WakeupOnAWTEvent)wakeup).getAWTEvent();
               if (events.length > 0 && events[events.length-1] instanceof
MouseEvent)
               {
                   MouseEvent event = (MouseEvent)events[events.length-1];
                   int id  = event.getID();
                   int mod = event.getModifiers();
                   Point3d closestVertexCoords;
                   PickResult[] pickResult2;
                   int type = -1;
                   if (event.getModifiers() == MouseEvent.BUTTON1_MASK)
type = FPPickEvent.POINT;
                   if (event.getModifiers() == MouseEvent.BUTTON3_MASK)
type = FPPickEvent.END_POINT;
                   if (type == -1) return;
                   
                   // The location of the mouse event.
                   x = event.getX();
                   y = event.getY();

                   // Pick the closest node
                   pickCanvas.setShapeLocation(x, y);
                   Point3d eyePos = pickCanvas.getStartPosition();
                   pickCanvas.setMode(PickCanvas.BOUNDS);
                   try
                   {
                       //pickResult = pickCanvas.pickClosest();
                       pickResult2 = pickCanvas.pickAllSorted();
                       for (int k = 0; k < pickResult2.length; k++)
                       {
                           if (pickResult2 != null && pickResult
[k].getNode(PickResult.PRIMITIVE) instanceof FAPObject)
                           {
                               //Yeah this is an existing Shape got to find
more shapes
                               FAPObject fapTemp = (FAPObject) pickResult
[k].getNode(PickResult.PRIMITIVE);
                               fireFPPicked(((FAPObject) pickResult
[k].getNode(PickResult.PRIMITIVE)).getID(), type);
                               wakeupOn(wakeupCriterion);
                               return;
                           }
                       }
                   }
                   catch (CapabilityNotSetException er)
                   {
                       er.printStackTrace();
                   }
                   
               }
           }
       }
       wakeupOn(wakeupCriterion);
   }
   
   
   public void addFPPickListener(FPPickListener l) {
       //FPPListenerList.add(l);
       fpp = l;
   }
   
   
   public void removeFPPickListener(FPPickListener l) {
       //FPPListenerList.remove(l);
       fpp = null;
   }
   
   protected void fireFPPicked(int fpID, int type)
   {
       /*if (FPPListenerList == null) return;
       FPPickListener fppl;
       Iterator it = FPPListenerList.iterator();
       while (it.hasNext())
       {
           fppl = (FPPickListener) it.next();
           FPPickEvent e = new FPPickEvent(this, fpID, type);
           fppl.fpPicked(e);
       }*/
       if (fpp != null)
       {
           FPPickEvent e = new FPPickEvent(this, fpID, type);
           fpp.fpPicked(e);
       }
   }
}

Hope that helps.

Cheers

Christian