Well, yes, but, you only have to move the old folder to the new one, for example, if your current tenant name is my-awesome-pbx and you want to rename to customer-pbx, then you have to do the following on the linux console:
mv /var/spool/asterisk/voicemail/my-awesome-pbx-voicemail /var/spool/asterisk/voicemail/customer-pbx
