if all the accounts will just have e-mail in their inbox, then you could run a script that empties the inbox folder of every user using zmmailbox. a scrdipt of something like this (run as zimbra user)
disclaimer: the following code is untested, written off the top of my head, and is probably not the most efficient way to do things, it's just meant to give you an idea of how to do stuff Code:
#!/bin/bash
for i in `zmprov gaa`
do
zmmailbox -z -m $i ef /Inbox
done