I haven't tried this, but if you use the toll-free access number for ATT, you may be able to listen to messages that way. Try calling that number from your iPhone; if it works, then you can put it into your phone book and dial it from the car.
https://www.customerservice.att.com/...l_accessvm_faq
UPDATE: that wasn't such a hot idea, because if you use the toll-free number, you then have to enter your mobile number and password. But if you call your own mobile number from your mobile, it appears you can get right to voicemail, just as you could with a non-iPhone. Try that. It worked on my 3G even though you said in your original post that it doesn't.