I use Google Voice for my main phone number. One of the features I like, but don’t use that often, is the ability to download voicemail messages to my computer. Those messages are also saved to Dropbox.
UPDATE: Google changed Voice’s interface. You’ll need to get back to its legacy version first, then follow these instructions. I’ve recorded a quick YouTube video on how to do that.
Here’s a graphic that shows how simple this is. When you go to Google Voice, click to display Voicemails. Then, next to the one you want to download, click the more drop-down option, Download, save the message as you normally would any file.