Welcome to the forums.
Yes, Zimbra will run on CentOS, I use it on CentOS 4.3 (I use the OSS version but the Network version is effectively the same) but the supported platform is RHEL. Because of the support status, I'd check with your support contact at Zimbra to see if CentOS is OK & supported.
As for Ubuntu, I think severeal people have got Zimbra working on it but again the question of support is your problem.