I brought the nerdy cube stuff today. It's a cube that's partially superglued together so you can only move two adjacent faces. I've had this for a while, and even though people talk about this particular subset of possible Rubik's cube moves from a mathematical sense, I haven't seen much in the way of discussion of how to solve it or its features. We'll declare the right face (yellow) and the down face (red) the two movable sides. Once both the URF corner and the URB corner are correctly placed, all of the corners are correctly placed. Once the three right face edges UR, RF, and RB are correctly placed and oriented, the remaining edges are all correctly oriented. So, the solution usually is:
1. Solve edge UR.
2. Solve the two corner-edge pairs (URF + RF) and (URB + RB).
3. Orient D corners (they are correctly placed automatically).
4. Permute D edges (the are correctly oriented automatically).
If there are any new-school cubers out there, you might make U and R the movable faces and solve DR first - upside down of what I just said. I originally started with the Nourse method, so I'm a little biased to R and D moves, even though it's a little harder to see what you're doing. Amazingly enough, this bias has overcome everything - even the fact that I'm a lefty who cubes right-handed.