I would think you just dial your own number to access your voice mail. Perhaps if that does work on the iPhone (It works on every other phone out there, even the dumb phones) just add a speed dial to your own number to #1.
I have visual voice mail on my phone. The only difference I think from Apple's is when I tap a selection, I get a full screen view of their picture, name, number, time and date they called as well as a transcript of what they said plus a rating system where I can rate the transcript based on how accurate it was (and so far its been very accurate) and I guess that gets sent back to the back end where they can improve the visual voice mail. Plus from there I can send a voice memo back, call them back speaker or delete.