1. Start iTunes from a fresh log-in. This means you need to log out of any other account. I normally toggle out of the "remember password" box as well.
2. After iTunes starts, go to the iTunes Store and select the United States store.
Careful--read the rest and follow the exact steps.
3. Go to the App Store and pick a free application. By selecting a free app, you can avoid selecting a payment method in a later step.
4. DO NOT LOG IN. CREATE A NEW ACCOUNT.
5. When the webform asks you to select a payment method, choose none. If you don't get "none" as a choice, leave iTunes, read these instructions once more, and start again.
6. Enter your information and offer an address in the USA. Google for 17603 or some other zip code and you'll find a proper address displayed.
7. Finish the registration and wait for the activation mail from Apple.
8. After activation you can log in with your US account and start downloading free apps.
Hat tip to Rutger Blom for reminding me to share.
iPhone Club offers information in Dutch.