Jonathan;

Try this with very small LEDs, dispersed in a pretty tight 3D grid and all-but-invisible driving conductors; the project is to animate a 3-D human figure walking/running along the inside edges of the cube ad-infinitum (or the off switch), turning in 90-degree angles when they reach an edge. The LEDs should not be so bright so as to prevent looking at or through them.

I assume that experimenters in 3-D displays go through such exercises.
 

Of course, you can always start with 3-D Pong...