Hey there! You may be wondering, how can this random forum-goer know Twisted Treeline's meta...Wellp I'm currently top 100 in 3v3s and I'm here to educate the public!
The strongest team composition at the moment includes the following:
1. A main jungler (someone highly mobile who can move in between lanes and make sure that dragon is captured everytime it spawns).
2. A second very strong champion who lanes and farms but can pick up a wriggles because it's imbalanced, giving vision of half the map and doubling the speed of dragon. These laners are very mobile and disruptive, usually can gank at early levels; and snowball
3. A strong laner. Preferably someone who is immovable from the lane, who has CC and can keep mid warded without dying.
The top main junglers go as follows: Lee sin, Udyr, Warwick(past 6), Tryndamere, Nocturne
The top top lane champions go as follows: Jax, Yorick, Gangplank, Singed, Riven, Renekton, Garen, Akali
The bottom lane champions go as follows: Karma, Rumble, Cho gath, Morgana, Poppy(yes she is crazy in this map but still has troubles farming) Yorick, Ryze, Lanewick, Garen, Gragas
^All in no particular order
WRIGGLES IS THE BEST 3v3 ITEM IN THE GAME.
It supplies wards all game, lifesteal, armor, damage, procs for taking down red, green, black, and dragon buffs.
Dragon is EVERYTHING. +280 gold and a global buff anyone? This gives you absurd gold and level advantages and makes the other team passive while the buff duration is in effect.
Vision wards and oracles are optimal on your beefiest champion to counter wriggles wards and ensure that you have constant vision of dragon and red.
Other popular warding spots are the bushes where jungle creeps are on the top of the map.
Twisted Treeline is different than Summoners Rift in the sense that things snowball incredibly fast. You aren't farming for a half an hour or stacking gold/10 to buy that big item 40 minutes in. You're going for early game advantages and quick disruptions to end the game.
Just because a champion isn't on the list doesn't mean it isn't viable. Those are just what I see as the most viable 3v3 champions based on my ranked games.