It's a bitch to actually try to move to any european country. I think the Czech republic has a limit of 3 months for anyone who isn't working.
If you are working...even if it's just over your computer running your business in the states - you still need to apply for a work permit and a residential permit as well as register with the local police.
http://www.expatriates.com/ has a lot of info
Mind you, if you are european you can dash from country to country as long as they are in the european union...must b nice.