I hope I'm not too late with this reply to help you. I just had the very same situation but they were messages from my husband who recently passed. I upgraded to the 5 but still had my 3G. The messages were on the 3G but were not transferred to the 5. I got absolutely NO HELP from AT&T or Apple.
But . . . I found this info on line, and followed the instructions and was able to save all of my dear husband's voicemails to my pc and now I've emailed them so I can listen to them on my phone as well.
You have to spend a little money but it only took me 1/2 hr at the most and I had all his messages safely back, so it was well worth it.
The info came from TechGeec on YouTube. It was excellent. I purchased DiskAid and FileApp. I didn't have the coupon code because I was much too upset so I paid $29.99. But I was able to retrieve all the Voice Mail and all my videos and photos that also did not restore to the iphone 5. The instructions were really simple, so simple I didn't think it'd work, but it did remarkably.
My heart was broken at the thought of not hearing his voice again. I know your's has to be as well. I really hope this helps you. If you have a moment, let me know if you try it. Wishing you well. All the best, donna