You can use the open standard CardDAV to synchronise your Zimbra contacts to your Android smartphone.
Android has no native CardDAV client but some are available from the Android Market. We have used the free CardDAV-Sync which seems to work well. We talk you through configuring that client but essentially any others that are available will be similar.
Once you install CardDAV-Sync it will appear in the Android Account Sync area, but simply starting CardDAV-Sync will give you the option to Go to accounts & Sync.
Select Add Account, and choose CardDAV as the account type.
The server name is eg zimbra.opensure.net.
Leave SSL ticked and then enter your email address and password.
The app will then check the connection and ask which address book to synchronize, select as required.
Contact details will now sync.