Red/black for 2D..what for 3D?

I am trying to find an algorithm that is an efficient way to update the 6 "neighbors" of a point in threespace.

Is this enough information to explain the problem? I suppose it is referred to as an n-body problem, where each n has 6 neighbors (cube).

What is the most efficient storage scheme for updating each n value based on it's neighbors, where every n must be iterated over. That is, the innermost loop will be forward and backward looking, in two directions. I have heard of 2D grids use a "red-black" scheme for this kind of update, but I am not sure what exists for 3D.

Thanks

Derek