I have put off this task for years because I thought it would be a lot harder than it really is. My voicemail was full today, so I was motivated.
I found some instructions that explain how to extract the voicemail files from an iTunes backup:
I was able to do the backup, make a copy of the backup folder on my desktop, and then use a script to extract the voicemail files to a separate folder. From there I can play them, rename them, etc. It makes use of the terminal window which low-tech users may not be familiar with. Anyone familiar with a terminal and command line should have no trouble.