QGIS is pretty nice.
Being so used to ArcGIS, I also find the uDig project to be very worth looking at: http://udig.refractions.net/
It also has a link to JGrass, which uses the uDig framework to provide a more usable GUI for Grass.
Grass is quite powerful - even more so than ArcMap in some regards - but can be a real pain to set up and get working, so I would try out the alternatives before going that route. But if you find you like GIS, it's definitely worth fiddling with over a weekend.
Unfortunately, I've not heard of ArcMap being successfully ran via wine, or else I would suggest trying that.
Also, if you're not strictly tied to a *nix environment, it may be worth checking with your university's Geography or Engineering department and asking if they have either 1) labs with ArcMap on them or 2) Student-use CD's so you can install ArcGIS on your home PC.
Even though ArcGIS is a $15,000+ suite, ESRI is extremely generous with universities, and gives it to them free of charge (including CD's for students to use).
Best of luck to you and your project - I hope you find a solution that works for you.