Would I need anything more than a passport in Poland, Lithuania, Latvia, Estonia, Romania, or Ukraine?

What country is your citizenship?

I'm sorry. The United States of America.

Not for Poland.

All countries except Ukraine are in EU, so you only need your passport. Except Ukraine and Romania they are also in Schengen, so covered by its 90-in-180 days rule and no passport control when travelling between them.
You no longer need a visa for Ukraine as US citizen - you get a stamp on entry allowing you a stay of 90 days. Your passport should be valid for 6 months beyond your proposed departure.