Downloading voicemail to my laptop
There is another way. Per this instructables an iTunes backup of the iPhone contains the voicemail files in these folders:


~/Library/Application Support/MobileSync/Backup/
Mac
\Documents and Settings\(username)\Application Data\Apple Computer\MobileSync\Backup\
Windows XP
\Users\(username)\AppData\Roaming\Apple Computer\MobileSync\Backup\
Vista and Windows 7


However, the bad news is that these voicemail files are mixed in with a ton of other random files from your iPhone, and all the files have long, meaningless filenames with no extensions. To determine what the files are, use the file * command. Note that the backup must not be encrypted for this approach to work.

file * | grep audio
file * | grep GSM


(on Windows install Cygwin and use the Cygwin terminal to get the file command)

The two file contents that seem likely are

RIFF (little-endian) data, WAVE audio, Microsoft PCM, 8 bit, mono 44100 Hz
Adaptive Multi-Rate Codec (GSM telephony)