A Hamiltonian circuit for Rubik's Cube has been constructed. Basically it is a sequence of quarter-turn moves that would (in theory) put a Rubik's cube through all of its 43,252,003,274,489,856,000 positions without repeating any of them, and then one more move restores the cube to the starting position. In fact, this Hamiltonian circuit only uses turns of five of the six face layers of the cube. The back layer is never turned.
This .zip archive (about 7 megabytes, expands to over 200 megabytes) contains a specification on how this Hamiltonian circuit is constructed. See the readme.txt file within the archive for detailed information.
Click here for more details about this solution.Correction (Feb. 21, 2012): My original .zip archive had an error in the "x.txt" file. The sequence for "x" was missing two moves. The corrected .zip file has a size of 7,096,554 bytes.
In December, 2011, on the speedsolving.com forum, I published a Hamiltonian circuit for the 2x2x2 cube (traversing all 3,674,160 positions using turns of the top, right, and front layers. This .zip archive (about a quarter megabyte, expands to less than 4 megabytes) contains a move-by-move list of this Hamiltonian circuit. See the readme.txt file within the archive for more information.