We show that in the limit where the number of spacetime dimensions D grows to infinity a very large class of black holes (including nonextremal, static, asymptotically flat ones, with any number of gauge-field charges, possibly coupled to dilatons) possess a universal set of quasinormal modes whose complex frequencies depend only on the horizon radius and no other black hole parameters. The damping ratio of these modes vanishes like D − 2 / 3 , so they are almost normal modes, or "quasiparticle" excitations of the black hole. The structure responsible for the existence of these modes at large D is also present very generally in other black holes.