Chad, I'm not American either, so my experience may not be that useful to you, but I haven't paid a penny in income tax to the UK govt for the past 7 years.
I spent 2 years working in the Bahamas. There is no income tax, true - but there is a (5%?) National Insurance levy. Plus, you need your own health insurance. Be warned, the health service in the Bahamas combines the worst aspects of both the British and US systems tgether.
All countries have to raise revenue somehow. The Commonwealth of the Bahamas is no exception. In their case, it is mainly through charging duty of up to 50% on items imported into the country. So, shopping at the local supermarket can be a bit of an eye-opener.
Also, you can't just go and decide that you would like to work and live there. It is sometimes possible for workers in certain specialist fields to get a work permit - provided there is no-one suitably qualified and experienced amongst the local population who can do the job. It is also possible to invest a significantly large amount into a business there and thereby gain residence.
I currently spend most of the year living and working in Kuwait. At the moment, there are no income taxes levied on expatriates, the cost of living is also fairly low.
However, again, you cannot just turn up. Expatriate workers must be sponsored in by a local employer.
So, yes, for some of us, it is possible to avoid paying income tax. But, as Gail has so eloquently pointed out, there are other more important questions you also need to ask if you want to avoid your tax burden by moving aboad.