Savings account for non US citizens

[Misja]

I have a fair amount of money on my poker account which I use as a buffer. Since most of that money is doing nothing most of the time, I would like to put some of it to a (US dollar) savings account. This way I could at least get some interest on it.

A couple of things I'm unsure about, here they are:
- Should I use a US bank one from another country? Which bank offers the lowest transfer costs?
- Should I be concerned about the US anti terrorism act when I use a US bank, even though I am European?

I'm very interested to hear any advice or personal experiences on this subject!