This is a very simple mod inspired by Oskar's carry a cube, which uses a 4x4 for overhang bandaging. This uses a simple 3x3 with the handle sculpted out of Milliput. The only obstacle, but an easy one, is to ensure that the middle layer can turn when at 45 degrees.
You can watch the video here: (it may take a bit time to process because I'm currently uploading it.) http://youtu.be/K4qM5iw5NN4

So the corners can't mix? It's a centers/edges only challenge?

Neat idea by the way. I'm imagining two cubes linked by this handle.

Something like this:
http://twistypuzzles.com/cgi-bin/puzzle.cgi?pkey=1584

No, I mean just two cubes made exactly like this, where the handles are intertwined.