There's an allowance for up to $70k or $80k where you avoid double taxation. I'm a US citizen with permanent residence in Germany and since my income has been below that all I have to do is fill out my return and not pay anything.
I think this only works for certain countries. It also applies only if you live outside the US for at least 330 days the first year, but those days don't have to be consecutive.
If you are from the US, you can never escape paying income tax unless you plan on never returning. You can't legally renounce your citizenship.