Ok, I found a plugin:
http://ww.wp.xz.cn/extend/plugins/widget-classes/
It was surprisingly hard to find. I had to make some changes to get it to work with my theme, but I’m happy with the result.
Oh my gosh it’s always something stupid and simple 🙂
My CSS files are in a css folder because I concatenate, minify and cache a master css file from several css component files. I’ll have to keep them in the css folder and write the minified version to style.css in the main template folder.
Thank you Snat!!