In ball-based sports it would make offense difficult as there'd be two teams on defense at any given time.
My buddy J. Daniels pointed this out to me, too. I still think it could work in a sport of the style like baseball or cricket, where the opportunity to instantly switch from defense to offense isn't there like it is in intercepting footballs and stealing basketballs.
We also banged out a hypothetical way to make tri-baseball. It's not anywhere close to perfect, but illustrates what I mean:
Take three teams: Nationals, Phillies, Mets. Each inning is divided into thirds instead of halves. In the top third, the Nationals are at bat, followed by the Phillies in the middle, and the Mets in the bottom (since we're playing at Shea, natch). Now, at this point, my brain said "So there's 18 defensive players on the field?" Mr. Daniels responded, "Poppycock," as he is known to from time to time. "The key is to split up the two defensive teams. It can't be an even split in baseball because there's nine defensive players, so the best you can do is 5 from one team, 4 from the other. So, split the field into two groups - in one, the outfielders and the battery, and in the second, the infielders. In the top of the inning, the Nationals are at bat, the Mets are in the outfield, pitching, and catching, and the Phillies are in the infield. In the middle of the inning, the Phillies are at bat, the Nationals are in the outfield, and the Mets are in the infield. In the bottom third, the Mets are batting, The Phils are in the outfield, and the Nats are in the infield."
There would be some interesting developments in that your outfield and infield would not practice together regularly, so would be out of sync to a degree. Rundowns when the runner is caught between third and home would be interesting, too.
Basically, I am now severly interested in inventing a sport involving three teams of players, that is logical, fun, and competitive.
(And CortJstr, you get a can for accurately predicting what a Redskins/Cowboys/Steelers score would look like, even if you've overestimated the abilities of the Cowboys...)