I myself haven't been to Prague yet, but elsewhere in Eastern Europe and the whole spirit of this area is worth getting to know. As for Prague itself, as you heard, it is at the moment a very popular destination, which means it will (if not yet) become soon artificial and touristy ! Go now, when it is still Czech like, not fully westernized.
If you have only three weeks to travel, I too would advise to keep it to two or three countries at the most.
There are plenty of threads on Eastern Europe/Prague in the «Where ya all going ?» forum.
Have fun !