Aren't there several Americans already living in Krakow?
The majority of Americans are most likely students here for study. Some for short term, some a little longer. There isn't a huge American population in Krakow. Pretty much everyone I've met here is staying for the short term. The American Embassy in Krakow doesn't seem to really know how many Americans live in Krakow or Poland. It's been fairly recent, Dec 2007, that Americans have needed any paperwork to stay here. So the statistics I'm assuming are pretty limited.
You can get a temporary visa for a year, if you take Polish classes for example. But you will need to renew that every year. After 2 years or 3, not sure, you can apply for a permanent residency. However you have to prove that you have money to live here and a reason to live here. You can't just decide today I want to live in Poland and move here. Polish lessons is the easiest, or family ties.