Thanks, dkarp for the suggestion.
I'm having trouble creating a script to do it, but I have figured out this much:
( either su zimbra or run zmmailbox in /opt/zimbra/bin/ )
zmmailbox -a username -p password [ use an administrator account ]
mbox> aa
admin@domain.com password [ again, use an admin account ]
mbox> sm
user_name@domain.com [ open the mailbox of a user ]
mbox> s "in:sent is before:-4weeks" [ do the search ]
I still haven't figured out how to delete them, or how to put the whole thing into a script.
Thanks for your help to get this far!