When staying in the same version, what I used to do was install Zimbra (same version) on the new server with the --softwareonly flag on install (to install the RPMs). Then you can (while the old server is still running and zimbra on the new one isn't running) do an rsync of /opt/zimbra and any other zimbra storage volumes/index volumes to the new server. You then stop Zimbra on the old server, do a final rsync, and start up on the new server.
I've used AJCody's wiki pages in the past for this:
Ajcody-Notes-Server-Move - Zimbra :: Wiki
Note: only works for same OS version and same Zimbra version.
State University of New York at New Paltz