I requested and received my unlock from AT&T yesterday, but they sent instructions requiring that I restore my phone with iTunes. This is nonsense, as I have no intention of losing my 7.1.2 jailbreak. I grabbed a T-Mobile SIM from a co-worker and tried the instructions below:

1. Turn iPhone off
2. Insert non original carrier SIM card (T-Mobile SIM in my case).
3. Connect your iPhone via USB cord to your PC or Mac and allow the computer to power on the phone.
4. Open iTunes on your PC or Mac iTunes may pause for a moment and/or show a pop up box with a failed activation message, click “ok”.
5. Briefly disconnect USB cord, then reconnect USB cord.
6. iTunes should now display a message saying “Congratulations, your iPhone has been unlocked”.
7. If prompted for a Carrier Update, please accept.

It appears my iPhone is now unlocked and usable with any active SIM card.