A trivial one that I've thought of but haven't implemented:
Take all of the "mobile" pieces. The visible parts (ignore the foot) form a convex solid with four sides - the two visible faces and the the two faces which are flush with the inner parts of the cube (just as a corner cubie has 3 visible and 3 inner faces). Grind the pieces down until they are essentially "flat", just two adjacent triangles with a fold in the middle and a foot behind them.
This changes the shape of the puzzle from a great stellated dodecahedron to a small stellated dodecahedron (see http://mathworld.wolfram.com/Dodecahedr ... tions.html
The complexity of the puzzle (presumably) remains unchanged.