Important: There is now a native version of Unity available for Linux! It's currently in an open beta. You can download it from here. It's already known to have great performance improvements and better functinality.
If you're looking for a working Unity setup, you should try the native version first.
This repository contains scripts to make Unity run on Wine the best it can.
If you are looking for instructions, you can find them here.
If you want to play Unity Web Player games on Linux, you should look at Pipelight.
If you want to give feedback, please do so here. Thanks!