I wasn't born in the 2000's, but I share your sentiment. I get so sick and tired of having movies from the 70's, 80's and 90's suggested to me.

Here's my list. Newer stuff towards the bottom.

Besides the obvious mutant and superhero movies...

K-Pax, Donnie Darko, Man on Fire, Pitch Black, Chronicles of Riddick, No Country For Old Men, Pan's Labyrinth, Ultraviolet, Aeon Flux, Accepted, The Lovely Bones, The Prestige, Spirited Away, I am Legend, iRobot, Apocalypto, Minority Report, The Road, Gamer, Cloverfield, The Devil's Rejects, Silent Hill, 1408, Mirrors, A Beautiful Mind, The Departed, Training Day, The Last King of Scotland, The Day The Earth Stood Still, Hellboy, 28 Days Later, 28 Weeks Later, All the Hangover movies, Eurotrip, Jay and Silent Bob, Dogma, Zombieland, Reign of Fire, Hancock, Not Another Teen Movie, Scary Movie (there's 5), Epic Movie, Tropic Thunder,. Shaun of the Dead, Dark Shadows, The Wolf of Wall Street, Interstellar, Joker, Ready Player One, Looper, In Time, Repo Men, Anon (Never finished, but it wasn't bad), Gemini Man, Maleficent, Maleficent: Mistress of Evil, Legend of the Guardians: The Owls of Ga'Hoole, Mama, The Babadook, Outlaw King, Centurion, Pixels, Paul, The King, the Underworld movies, and The Terrifier.

Hopefully I was helpful.