I cobbled together a really simple little Firefox extension that periodically checks the server and puts a notification in the statusbar if there is new mail. The notification is clickable: it brings up a popup menu with the titles of the first 10 new messages in your account -- and if you're runnining v4.5 or later -- choosing an entry from the menu can take you directly to the message in the HTML client (unfortunately there's no direct-to-message URL support for the Ajax client, so clicking on a menu entry just takes you to the client).
You can install it from: http://curple.com/zimbracheck/.
Known Issue: the extension doesn't handle auth token expiration correctly, and so once yours expires (after ~24 hours by default I think) you will need to quit and restart the browser.
Oh yah, and just to be ultra-clear: this is something I did on my own time and is completely unsupported by Zimbra. So don't go asking the support people for help with it -- they'll just come yelling at me, and then I'll have to remove the download and nobody will be able to use it.