I just helped a friend in the US to transfer from her iPhone 4 to iPhone 6 plus and had the same issue.
We did an iTunes back-up and everything transferred except voicemails. She had 3 years worth of important voicemails that she wanted to keep.
She is with ATT and apparently they only keep voicemail for 2 weeks then it is automatically deleted from their server.
Luckily she had backed up her old phone to iTunes AND that was backed up in Time Machine!
If you backed up your old phone in iTunes you can retrieve the voicemails from there but they are not easy to access - we used 3rd party software to retrieve them as described in this article: http://www.macroplant.com/iexplorer/tutorials/how-to-access-voicemail-on-iphone
IMPORTANT - do NOT back-up your new phone in iTunes until you have retrieved your voicemail.
iTunes will overwrite the back up of your old phone with the info. on your new phone - then the voicemails will disappear.
We did this 😟
but because my friend had TIME MACHINE backups, we were able to enter time machine and copy the old iPhone backup into the backup folder on her MacBook:
Library > Application Support> MobileSync> Backup
Then we could install the iExplore software and she now has access to her voicemail again.
::phew!::