I'll just list a few off the top of my head depending what you're looking for and I'm not sure which are native linux:
Dystopian setting with some story? Dues Ex series, Bioshock series, System Shock series or Spec Ops: The Line
Loot fests or weird stories/characters? Borderlands series
Survival fetish? Any of the S.T.A.L.K.E.R. games.
Story and puzzles? Half-Life 1 & 2
Old school run and gun? Wolfenstein: The New Order
Horror atmosphere? Metro series or F.E.A.R. series
Destroy shit? Red Faction: Guerrilla (Mostly third person)
Campy 80s movie style? Far Cry: Blood Dragon
Strong stealth elements? Hitman series, Splinter Cell series, Dishonored & Thief 1 & 2 (Some may be third person)
More simulation side? ARMA
Adventure? Tomb Raider (2013)
Tactical Shooter? Tom Clancy's Rainbow Six, Rainbow Six Vegas 1 & 2, Ghost Recon & Advanced Warfighter 1 & 2
Some may be a little old, but may have mods to help with the graphics.