Season 1: Sokka (the Cynic), Katara (the Optimist), Aang (the Realist — ironically the mediator most of the time while the siblings argue), and Zuko or Momo/Appa (the Apathetic).
Season 2: Toph (the Cynic), Katara (the Optimist), Aang (the Realist), and Sokka (the Apathetic, or a milder Realist). Alternately, for the dark version which fits it even better: Zuko (the Cynic), Ty Lee (the Optimist), Azula (the Realist), and Mai (the Apathetic).
Season 3: Aang (the Realist), Katara (the Optimist), Sokka (the Cynic), Toph (the Apathetic), Suki (another Apathetic), and Zuko (another Cynic).
The Four Nations as a whole fit. The Fire Nation is The Cynic (due to thinking less of everyone else), the Earth Kingdom is The Optimistand the Realist (they're the main obstacle for the fire nation, but they also think hiding behind a wall will protect them from anything the Fire Nation throws at them), the Air Nomads were The Apathetic (they were the most detached from the world), and the Northern Water Tribe is The Conflicted (they want to help out, but they don't have the resources and are too isolated to do much).
The Belchers from Bob's Burgers: Linda is the Optimist, Louise is the Cynic, Bob is the Realist, Gene is Apathetic, and Tina is Conflicted.
Codename: Kids Next Door: Numbuh One - Nigel Uno (the Cynic), Numbuh Two - Hoagie P. Gilligan Jr. (the Optimist), Numbuh Three - Kuki Sanban (the Apathetic), Numbuh Four - Wallabee Beetles (the Cynic), and Numbuh Five - Abigail Lincoln (the Realist).
Numbuh One potentially fits better as The Conflicted, shifting constantly from optimistic (almost dangerously so) to cynical depending on the circumstances.
Ed, Edd n Eddy: Ed is the Apathetic, Double-D is the Optimist, and Eddy is the Cynic.
The Powerpuff Girls: Blossom (the Realist), Bubbles (the Optimist), Buttercup (the Cynic), Professor Utonium (the Apathetic)
Rex The Runt: Rex (the Realist), Wendy (the Cynic), Bad Bob (the Optimist), Vince (Apathetic).
Rugrats: Tommy (the Optimist), Chuckie (the Realist), Phil and Lil (Apathetic), and Angelica (the Cynic).
Scooby-Doo! Mystery Incorporated: Fred (the Optimist), Velma (the Realist), Daphne (the Apathetic), and Shaggy (the Conflicted). The role of the Cynic is filled by various secondary character. Most notably Sheriff Stone. Interesting to note is that while they fit this pattern both philosophically and functionally Fred, Velma and Daphne have personalities more closely associated with Apathetic, Cynic and Optimist respectively.
The Simpsons: Homer and Bart (the Cynics), Marge (the Optimist), Lisa (the Realist), and Maggie (the Apathetic).
South Park: Stan (the Realist), Kyle (the Optimist), Cartman (the Cynic), Kenny (the Apathetic), and Butters (the Conflicted).
Teen Titans: Robin (the Realist), Starfire, Beast Boy and, to some extent, Cyborg (Optimists), and Raven (both the Cynic and the Apathetic).
Teenage Mutant Ninja Turtles: Leonardo is disciplined, balanced and the group leader (Apathetic, yet the Realist); Donatello is intelligent, has special skills and always busy (Conflicted, yet the Apathetic); Raphael is hostile, skeptical and rude (the Cynic); Michaelangelo is foolish, determined and optimistic - (obviously the Optimist).
ThunderCats (2011): Lion-O and Wily Kit (the Optimists), Tygra (the Cynic), Cheetara (the Apathetic), Panthro (the Realist), Wily Kat (both Realist and Apathetic), and Snarf (the Conflicted).
From VeggieTales: Bob the Tomato (the Realist), Larry the Cucumber (the Optimist), Archibald Asparagus (the Cynic), and Jimmy and Jerry Gourd (both the Apathetic).
Young Justice: At the beginning when there were only 4 members, it was Speedy/Red Arrow (the Cynic), Kid Flash (the Optimist), Aqualad (the Realist), and Robin (Apathetic). Later, when Speedy/Red Arrow left and Superboy joined them, he replaced Speedy as the Cynic and Miss Martian came in as the Conflicted.
Robin is sort of an odd duck in that he fits the apathetic definition most of the time, but this is at least partly because he doesn't like to slow down long enough to actually consult with his team members much less debate philosophy. Whenever he actually does slow down to talk strategy with the rest of the team, it's usually because the situation has gotten so bad that Robin has taken over as de facto commander at which point he doesn't debate, he just gives orders which the others meekly follow.