Unfortunately, this is Not and error with your bluetooth device. Currently, (and why I Apple did this is beyond anyone but them) the iPhone's voicemail does NOT work with bluetooth. I discovered this when I was trying to get the voicemail to work over my non-apple bluetooth headset. (The apple bluetooth works for voicemail without having to do this.) I have found one trick, however, that does work with the headsets but not with my Durango's bluetooth.
1. Connect the bluetooth device to your iphone ( you should see the bluetooth symbol lit up at the top left of your phone next to your battery strength indicator.
2. Navigate to your visual voicemail location of the phone
3. In the top right hand corner you should see a button reading "Audio". (If it says speaker then your bluetooth is not connected.)
4. Press the "Audio" button.
5. You should now see a list of different audio choices - iphone, speaker, and [bluetooth device name]
6. Select your bluetooth's name from the list.
7. Now play one of your voicemail messages and you should hear it over your bluetooth headpiece.
This does not work with MY car's bluetooth but I have heard it works for others. I did get it to work with my non-apple bluetooth headset but finally stopped fighting and bought the Apple one.
Upside: THIS WORKS FOR MUSIC and PODCASTS ALSO! So now you can have your bluetooth earpiece in and listen to a podcast or music while you walking or driving. Note that it will run your battery down faster though.
Downsides to this method are:
1. Manual process (apple bluetooth it is automatic so you don't have to do all that.)
2. Your voicemail and music will play over your external iphone speaker while it is broadcasting to your bluetooth, so make sure you don't mind if other people hear it!
3. You have to reset this after EVERY phone call, in coming or out going. (This one is the reason I finally gave up and bought the Apple bluetooth.)
Try this method for your car and see if it works for you. Doesn't for my car but I know it does for some. I got this tip from the best iPhone podcast on the internet - Today in iphone. Look it up and check it out. It is worth downloading all of his podcasts from the very beginning and working your way through all the tips because you will find all kind of new things.