If you're looking for PvP opportunities beyond the usual battleground and arena scene, the upcoming Tournament of Champions 2.0 may be a perfect fit. This player-run PvP tournament is 3v3, single-elimination style (with double-elimination in the semi-final rounds). If you want to enjoy the fun -- and have a current rating over 2200 -- you can sign up through February 15th with the tournament tentatively scheduled to start on the 21st and last for two to three days.
Just interested in watching some great 3v3 teams compete? The tournament will be streamed on Twitch for everyone to enjoy. For full details hop over to the forums.