Visual Voicemail is not an iPhone only feature, because a company called Klausner has the patents on it and licenses it to Apple and others. Which is a good thing as Visual Voicemail is a great feature and I would not want to be without it again.
BoyGeniusReport reports that AT&T will update the BlackBerry Bold on Tuesday to enable Visual Voicemail on this BlackBerry smartphone. The report also says that one would be able to get AT&T to enable the feature before Tuesday.
As calling customer support is a pain, I would just wait until Tuesday and hope the upgrade is easy enough.
The Visual Voicemail for BlackBerry is supposed to offer up to 14 days of storage for 40 voicemails, forward voicemails as an MMS and archive voicemails to device memory or on a microSD.

Shop for the BlackBerry Bold on Amazon.com.