Since I always have my thunderbird running and I really love a clean desktop, I want my thunderbird accessible in tray.

You have different tools for doing this, but there aren't to many that work on my linux box.

I found a few solutions. After testing these, there was only one left:

moztraybiff. It has one downside: it disables the new message pop-ups, but they are working on it.

Just download, install it and off you go. If you want to change the preferences, you have to go to the add-ons window.