A lot of users are reporting that their bluetooth devices are having issues working properly with the iPhone 5 (which has bluetooth 4.0 as opposed to bluetooth 2.1 which was what the iPhone 4 and 4s had). Here are some steps to try and fix these issues that are plaguing iOS 6 users.
Step 1) Under Settings > Bluetooth, first try toggling the On/Off switch (yes, this is trivial, but this has fixed a few problems)
Step 2) Again, under Settings > Bluetooth, remove the device that is having issues and then re-add it. Sometimes this can be an issue because you restored your iPhone from iOS 5 and the bluetooth settings didn't translate through the upgrade.
Step 3) Make your other device (not running iOS 6) attempt to pair first, and keep your iPhone's bluetooth off. Then, once the other device has sent out it's request signal, turn on your iPhone's bluetooth. Sometimes it runs into problems where it is unable to search around for a new bluetooth device and this step attempts to fix that issue.
Step 4) Lastly, make sure the device you are trying to connect to is compatible with bluetooth 4.0. Like I said before, Apple introduced using the newer standard in the iPhone 5 and caused many devices that only worked with bluetooth 2.1, to not function correctly with the iPhone 5. If this is the case, sorry, you need to buy a new device (or if it's a car, look into buying a new receiver).
Let me know if these steps helped you in the comments below!