Is there a way to copy digital voice mail messages to audio files that can be saved on a computer or burned to a CD?

I tried the little gadget with a suction cup on one end and a plug on the other that can be fed to a cassette recorder or computer audio input, but it didn't work.   It didn't pick up the audio from the voice mails.

The voice mail I have is via my phone carrier, so it's not stored in any device at my home. I want to save some of the voice mail messages and drop the voicemail service that Qwest offers.

Is there any way to run a plain telephone cord from the phone (plugging it in where the receiver plugs in) to the telephone input (the one I used for faxing) on my computer and copy the voice mail messages that way?   


If you have any experience with this sort of thing or knowledge of techniques that work, please let me know.

Thanks,

Genie