This is really well done! I love the background music, and it's an interesting puzzle mechanic I've never seen before. It took me a minute to realize that it is a turn-based puzzle game; at first I was trying to rush through the level quickly. But the dots that show where you're going to go made it pretty clear.
I think you should add a level counter on the screen somewhere, so I can say things like "I'm stuck on level 3."
...By the way, is level 3 solvable? It seems impossible to me!