In case no one's mentioned it yet:
Eidswick, J. A., "Cubelike Puzzles -- What Are They
and How Do You Solve Them?", 'American Mathematical
Monthly', Vol. 93, #3, March 1986, pp. 157-176.
From the introductory section:
"This article is an attempt to put some algebraic order
into the business of solving cubelike puzzles. Ideally,
an algebraic theory would unfold that would, in an
elementary way, yield highly efficient algorithms for all
such puzzles. The work here is a step in that direction."
Included in the applications section are the tetrahedron, octahedron,
dodecahedron, icosahedron, and the general n X n X n cube.