I kind of agree with Jimmy about maybe fitting in too much in 3 weeks. I did the same thing the first time I went to Europe, it's hard to resist wanting to see and do everything.
You may want to just run the numbers and see just how much time you would be spending on the train. The DB website is probably the best for that, it has train schedules for all of Europe.
http://reiseauskunft.bahn.de/bin/que...rotocol=http:&
If I had to eliminate one city from your itinerary, it would probably be Vienna. It's a nice city, and the Schonbrunn palace is pretty amazing but you might be architectured out after Rome. Or you could always do a side trip to the Versailles Palace from Paris. Vienna is also the one city that seems kind of out of the way on the map if you're doing a loop, it can almost be considered Eastern Europe, geographically.
Amsterdam has really cheap flights to Rome on EasyJet ($70-$100). So you could check out Amsterdam, fly to Rome, and then work your way back. Taking the train both ways across the continent could get tiresome.
Maybe Rome to Nice by train, and then fly from Nice to Paris ($49 on EasyJet). And then Paris to Munich by train, Munich to Amsterdam by train. Whoops, forgot about Venice...!