I've got a user on Windows XPsp3 using Firefox 3, IE7 and Chrome, depending on the situation. And all three of these exhibit the same behavior.
She fires up Zimbra (my server is v5.0.11, RHEL5x64), goes to another window to do some other work or compose a new message (which opens in a new window, btw), and while she's working, if she we're to get a new message in her Inbox, that browser window will obtrusively come to the front of all the others, as opposed to just blinking or making a noise.
This happens across every browser.
...so I installed Zimbra Desktop for her as a workaround until I could figure this out, and, much to my dismay, this behavior persists in Zimbra Desktop!
Her account is the only one this is happening to, and if I log in as her it doesn't happen to me on this machine. I'm totally at a loss as to what might be happening.
Anyone got a clue?