Solution: There are two things you can do:
1) Turn your phone all the off (hold down the power button for several seconds, then "slide to power off"), then turn it on again. If your Visual Voicemail used to work and stopped without reason, this will probably fix the problem.
2) Call your own phone number and access Voicemail the old fashioned way.
Problem solved: power cycle your phone
Workaround: call your own number to get into Voicemail