I root for riders. A rider's style, aggression, emotion, smarts, sheer power, whatever.
The only team I can ever recall rooting for is Quickstep, in all it's various incarnations. They used the team concept to full advantage so often. Great at recruiting and finding riders, great at getting the most out of them, great at tactics in one-day races. I really do enjoy one-day races more than stage races, they're so much more dynamic and tactical. GT team tactics are fairly tied into who's the strongest. There are teams I've not liked in the past because of boring tactics and shady behavior, but the riders also played a part in my dislike.