I know every time there's a post like this, somebody says "use Google Voice," but that is really your best option. Sign up for it and install the app on your phone. You don't need to import a number or use it for everything. You can just set it up to handle your voicemail. That way, your voicemail will be available through any browser and you can save it and relabel it.
As for the ones you already have, I'm not sure they can be downloaded as a file and moved around. There might be an app in the market to help, but customer service should give you some answers.