Oh, woops! Misread TextEdit for TextMate! That said, I find that Wordpad is a fantastic simplistic tool for creating school papers. For Linux I tend to use the slightly more complex (yet still very clean) LyX.
I've never used LyX; forgive me if this is a dumb question, but is it available for Ubuntu? I might boot up a virtual machine and give it a try if so.
It's also available for Windows/OS X so you might be better off just trying it on your primary OS.
I don't think you're going to find the kind of simplicity you enjoy in TextEdit though.
Personally I use text-editors e.g. vim, for every kind of writing I do, code or not.
Plaintext formats like http://eleanor.rubyforge.org/ (for screenplays) and Taskpaper (for planning/todo lists) are really helpful for this sort of thing.