I too have been experiencing this issue. It began after I downloaded the newest IOS 7.0.4 (after it's release, i waited a month to download. My experience has ALWAYS been that, after downloading most current IOS, I have problems with my phone.)
Not only has the visual voicemail been unavailable, I also have trouble connecting to wifi. My phone keeps dropping the connection. If I turn the phone off then back on, wifi is available for a short period of time, then it drops again. If i turn off Cellular Data, the wifi picks up and remains connected. However, when I turn Cellular Data back on, my visual voicemail remains unavailable until i reboot the phone.
Inconsistencies like this, with clumsy work-arounds, are a compelling reason to swithch to another make of phone..