Really fun and imaginative strategic game; I can see this being an interesting board game as well.
As many people have mentioned, not being able to see what your number is going to be after you move was a big obstacle in acting strategically. I ended up getting a real d6 and turning it over on my table in unison so I would know! Maybe like a confirmation would be a simple way to implement this? I.e. after you move, you can either press space to confirm that move, or another button to undo. That way you don't have to deal with the issues of showing a 3D object in a 2D game, but still give the benefit of adding to the strategic element.
Anyway, great game; very fun!