Imagine the possibilities. 3 teams of 3, fighting on a map catered to the game mode. 2 allies and 6 opponents, but with 3 of those opponents also opponents to the other 3. It'd be along the same levels of craziness as URF, without destroying the game's mechanics!
Hell, Dominion's map shape would be perfect for a 3-team brawl, and Dominion's gameplay style adds up well with the competitiveness of a 3-team match. In a map where taking objectives and kills is everything, having an additional team pit against both you AND your opponent would make for an awesome match.
Team up to take down powerful opponents, only to turn on them to further your own kill count. Not only that, but being able to go against 6 opponents would allow you to get a Hexakill, so the introduction of such a mode would allow for the Hexakill game mode to spread its horizons. Hopefully Riot takes it up one day.