I have tried this method on two different iPhones while using different computers and AppleID's. Here is the method I used to successfully unlock the iPhone without upgrading to iOS 7.
Both phones were the iPhone 4 model.
Both phones were using iOS 6.x.x
I called my provider and asked for the phones to be unlocked.
They told me that the IMEI was entered into Apple's servers and I just need to connect to iTunes to unlock.
Both unlocks were done through iTunes, one on an iMac, the other on a MacBook Pro.
Step 1: Backup the iPhone using iCloud or iTunes as desired
Step 2: On the iPhone, go to Settings:General:Reset:Reset All Contents and Settings
Step 3: Allow iPhone to complete the reset and wait for it to restart
Step 4: Plug iPhone into computer and wait for iTunes to start (or start it manually)
Step 5: Message will pop up to either Set up as New Phone or Restore from Backup
- at this point I was expecting the "Congratulations you are Unlocked" message, but it did not appear so I disconnected the cord to the computer and continued.
Step 6: Disconnect Sync cable, on the iPhone, go through basic setup... skip any steps it will allow you to skip.
Step 7: Go to Settings:General:Reset:Reset All Contents and Settings
Step 8: Allow Phone to reset and restart but do nothing else on the phone
Step 9: Connect to iTunes again
Step10: Message appeared, "Congratulations, Your phone is now Unlocked"
Step11: Restore your previous backup from iCloud or iTunes
This method worked for me more than once. Both times I had to connect the iPhone to iTunes twice in order to get the unlock message.
Hope this helps.
Original message: in response to apple-id