No! You would not have to pay American taxes on the income you have earned while living abroad. There is the foreign income exemption which essentially says that the IRS does not care to tax the first $100,000 you earn. For taxes above that, you first deduct dollar for dollar what you have paid to the foreign tax authority. Ball-parking it, you would have to earn more than $180,000 per year to need to pay US taxes. You still have to file US taxes, which is a cost and an inconvenience, but is manageable.

Of course, you must be genuinely living abroad to apply for this. This means either spending enough time outside of the US (> 330 days in a tax year, if I recall correctly) or have enough proof that you are really living abroad (things like primary residence in your name, bills, bank accounts, mortgage, insurance, kids at school -- regular every day stuff).

Now if you want to game the system and just "kind of" live abroad in order to take advantage of the income exemption, that's a different story.