All right!!!! I got it, my second time up here...lemme think...Ill come back and edit this with my riddle. Untill then...stare lifelessly at this space.
BTW when u said 3rd cycle i thouhgt u ment the third cycle of the games, which got me guessing R/S Pokés. The presidential thing was the kicker tho. Didnt really get the predecessor thing until u said it just now.
EDIT!!!
Ok, here we go:
1. TV's "____ Mice from Mars" used it versus Ash and Co.
2. "Eye of the Tiger"
3. "Digimon," "Pokémon," and "Monster Rancher"