Swidd
You will need at least Permanent Resident status (commonly known as a "Green Card") which is obtainable through marriage or occasionally employment if your role is one which requires special skills/qualifications that are not typically held by a native e.g. a JAA qualified examiner employed by a JAA approved facility in the U.S. creating local jobs & feeding the economy.
You are probably already aware that there are jobs in the regionals right now but pay & lifestyle are poor for the first couple of years. You will be subject to additional background checks than a native as a result of 9/11 which may slow down the process a bit. I know a few non-citizens who fly professionally in the U.S. however I can't think of any who were hired post 2001 (not to say none have, I just don't know any).