There's a fairly simple logical crutch that can be converted in to math I don't understand to prove it can't be done elegantly.

The first step is to read Nate's second post, which is true even though it isn't proof.

The second step is to assume that any solution worth looking in to requires mirroring the dominoes around the center of the cube - that is to say that for every piece you place, you place another one 180 degrees around the center of the cube from that tile.

Given this assumption, there are only two elegant ways to fill the cube: start in the center and work out - which gives you two large L shapes around the perimiter with an odd number of squares, and by starting in the center, which gives you Nate's first example.

Therefore, it's impossible to fill it elegantly