I am actually using Ubuntu 13.10, but this solution should work fine at least back to 12.04. Firetray is an extension for Firefox that makes it so you can minimize to tray on close, and minimize on startup (you'll see the Thunderbird window popup for a quick second, but it's hardly an issue). Then just add thunderbird to Startup Applications and when you login thunderbird will flash for a second then be minimized in your system tray. It also has full support for the default messaging menu so it doesn't create a secondary thunderbird icon.
Now for those that may have tried this in the past, I know that I tried Firetray a couple years ago and it wouldn't work at all, it had lots of bugs when used with modern Ubuntu, but the latest version seems to work flawlessly with Ubuntu (at least version 13.10, but I don't see why it wouldn't work with any other version).