Unless you're going to mount the cube on its corner or hang it from something then it makes sense to me that you wouldn't want the bottom face to move. In that case it seems the mechanism could be pretty simple. Just put the motor at the bottom and have it drive a vertical axle that has a lift arm attached that's long enough to push out a side but short enough to be able to go around without hitting the cube edges. You could have multiple if you want to push more than one wall at a time. Basically a very simple take on the music box idea and you only need the top half of it. Use a ramp rotating on top of the axle to drive a vertical pusher for the top cube face. Since you have your very clever net construction your walls should already be springy, you just need to poke them with something periodically.
Seems like it should work but I'm sure easier said than done. Also it would be very regular. You'd need multiple layers of pushers more like the music box above if you wanted it to look more random. Or maybe a coupler layers turning at different speeds on the same axle using a construction like a clock.