We've seen this type of game many times before. See list below.
What's particularly good about Move: A Brain Shifting Game? The easy levels are super easy. I did about 100 levels in 25 minutes. The graphics are very good. There's almost no delay between puzzle levels. Little ads pop up, but they are a minor burden.
I don't know who WarrenO is, but he solved all 1200 levels and rated it 5 stars on the App Store.
I did not buy the full version but I did sample 10 levels from the hardest levels. They didn't seem too hard.
Similar games (this list is incomplete):
Movez Pretty much the same as Block Mania but it has 1680 levels. I've solved 592 of them. That's probably enough.
Frozen Ice Cubes What's different: there's a point of 'no return'.
Escapology, reviewed by Nicola Salmoria. He told me about this game ages ago. I played it and enjoyed it, but never blogged about it.
Block Mania Similar to Movez
Move Puzzle - A Funny Strategy Game, Matching Tiles Within Finite Moves
Vubu What's different: the vubu men have different powers. Very sophisticated, very intelligent. Possibly the very best in this genre.