Use Zimbra? This add-on notifies you when new messages arrive in your Zimbra mailbox. You can view your next appointments and see also ongoing tasks.
Here are some more details about this add-on from the developers:
Zimbra Mail Notifier checks your Zimbra webmail account and notifies the number of unread messages. When new message arrive in your mailbox, a system notification is posted. You can view your next appointments (save in your Zimbra calendar) and be notified by the system. Tracking ongoing tasks is also available.
Learn more about the Zimbra Mail Notifier on the Firefox Add-ons website.