AT&T allows new customers to sign-up for any of their calling plans (not pre-paid) at the listed rate (sometimes these plans are referred to as "sim-only" since you will need to purchase an AT&T sim card if you don't already have one). You will not get the discounted subsidized price for one of their phones, but for a lot of people, they either have their own unlocked phone or would rather buy a used one off ebay or just dislike the idea of being locked into a 2 year contract.
You can do this over the phone or in person at an AT&T store. I suggest going in-store since a large percentage of AT&T reps have no idea that this can be done so there is a bit of confusion. You will pay a little more for the sim card, $26 (if you don't have one) vs over the phone, $10, but the extra cash is worth it so you can go back and talk to someone in case you have problems with your first bill, which you probably will.
Therefore, these are in-store instructions.
1) Go to local AT&T store, not a reseller like Best Buy/Costco/Circuit City
2) Bring the phone you'd like to activate with you
3) Pick the plan you want and prepare to be redirected by the sales rep to pre-paid plans or flat out being told that it is not allowed.
4) Stick you your guns, ask for a manager. If the manager says you cannot, have them call it in. At this point the tide turns and it seems that the store employees find out for the first time that you can indeed get a regular calling plan with no contract.
5) Wait. It took about 45 minutes to go through the whole process. Not only had the AT&T in-store employees never heard of this, but apparently their computers aren't setup to handle signing up with no contract. I left the store with printed contracts that had a lot of info crossed out an initialled by the Manager. A correct hard copy was created at AT&T HQ and faxed to the store which I came back a few days later and picked up.
6) Wait for first bill and make sure charges are correct. If there are any extras, go back to AT&T store and make sure you dea with the same rep that worked with you the first time.
7) Enjoy being contract free.
Apparently non-contract plans are available with other carriers, but this article is specific to the AT&T experience.