Well, I definitely ran 9V through the accelerometer and blew the thing.
That's okay though, because science.
Anyways, I realize now what I really need is a gyroscope. If I can get the instantaneous angular velocity and mount it on a fixed point I can use the parallel-axis theorem to use the space diagonal of the cube in order to get an arc-length velocity in meters per second.
That was quite the mouthfull, wasn't it?