I went to thrift store today. Didn't find any Scrabble. But, I did find some 1.5" square blocks. Ten for $.99. Enough for a 2x2 cube. And when I find more blocks I can reuse those as the corners of a 3x3 cube.
I can use the cubes as is. 2x2 seems to stay together fairly well and it's easy to rotate four blocks. I'd like to drill holes and add magnets to the inner faces so they stuck together. Think that would be needed for a 3x3. Wonder if I have the skill, I certainly lack the drill press/router that would make it easy. [Note: on my skill. Middle school shop class, I picked the very most easiest project. A box, to hold magazines (Dragon's actually). When I set it down on teachers desk to be graded it split at the seems and fell apart.]
Your dungeons will be assimilated!
This is an interesting project. Due to nature of 2x2 Rubik's Cubes three maps will always be adjacent and it's worth some consideration in choosing the maps and alignments. For instance I put all the natural cave tiles on two cubes.
Next up, besides shellac, is the keyed Dungeon of Geometric Doom and rules on when/why cube rotates!