Can’t recall where I found this but GotVoice is a new, free service that can pick up your voicemail and email it to you, optionally deleting the voicemail in the process. Right now it can do cellular voicemail and home voicemail (not answering machine but the voicemail service you can buy from your home phone service provider). So far, I’ve tried out the cellular but not the home and it works just fine. It can check and collect voicemail from several numbers and I don’t (yet) know if there’s a limit to how many you can have.
It can check your voicemail on a regular schedule or only on demand when you log in to their web interface and explicitly request it. Your schedule options are pretty limited right now. Standard schedule checks at 9AM and 2PM on weekdays. You can change that to occur every day or you can define your own schedule, picking from daily or only weekdays and choosing 3 times that your voicemail is checked. You can also optionally have your voicemail deleted daily or only on Sunday night.