Hexen II is a colourful adaption of the Quake engine, and in terms of source code and theme is much closer than Quake II to the original game. Unfortunately, Raven Software released the game with many rough edges... So beautiful, yet so cruel.
Hammer of Thyrion is the main Linux Hexen II port , and has had much work done towards squishing bugs and enhancing the OpenGL graphics. While it is not as widely distributed as Quake, the HoT demo includes some of the game's best levels and is available from the Sourceforge project page.
Quake II is the sci-fi themed sequel to Quake.
Early Linux releases weren't very solid, especially the mouse handling, but there is now a few related projects to choose from. QuDos' Quake II is the most recent, being based on Icculus Quake II, but having an enhanced graphics engine. Michael Olson has some Quake II source and binary rpms which appear to be old Icculus releases. Alternatively, there is the QuakeForge Quake II project which also provides support for multiple operating systems.
The multiplayer paintball mod Digital Paint 2 is based on Quake II.
See the old version of this how-to for Quake II troubleshooting tips.
The third Quake installment was a landmark multiplayer game, with some of the most beautiful and well balanced fragging ever. It was one of the first games to receive a full Linux commercial release.
Recently ID Software released the source code, and you'll find an Open Source Quake III project at Icculus.org.
Most mods should work. The Linux Gamers FAQ reports:
Yes, ... modifications work in Linux as long as they are compiled to the Quake III ... VM bytecode as advised by id software. ... modifications compiled to Windows library files will not work.
The retail add-on Quake III Team Arena is supported and, despite never getting rave reviews, is a great game.
For an atmospheric Q3 single player conversion visit The Dark Conjunction.
The Zerowing installation and known issues guides hosted by ID Software.
Linuxgamer's Quake III Howto.
Linux Question's Quake III forums for Sound and Mouse problems.
All things Quake III at Planet Quake
Quake IV is an epic FPS from the awesome Raven Software. It is based on the Doom-III engine and has high hardware specs. A 2ghz cpu and 512meg ram are minimum requirements.
Linux Questions Quake IV bug forums