A better question to me would be "What games do you play that aren't from before the turn of the millenium?"

Every old Final Fantasy still gets the odd play through. I have countless NES and SNES roms, as well as the original Lunar Silver Star Story. Starcraft and Warcraft II still get played offline occasionally by me.

Day of the Tentacle never gets old, kids ;D

Chrono Trigger has me still trying to get all the endings.

I also play lots of freeware, alot of which from the past few years, but it's all very ''old'' looking.