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 notable. It is based on Icculus Quake II, and includes some beautiful graphical enhancements - including cell shading - and support for FreeBSD and XMMS plugins. Another project still in development is Quetoo which has speed and security enhancements, but also a very stripped back interface and a few bugs.
Older projects include Michael Olson's source and binary rpms (which appear to be based on an old Icculus release) and the QuakeForge Quake II project with support for multiple operating systems. On a slightly different note - the Jake2 project is a version of Quake II written in Java.
The multiplayer paintball mod Digital Paint 2 is also 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, and check out the alien themed Tremulous for an original multiplayer mod.
The Zerowing installation and known issues guides hosted by ID Software.
Linuxgamer's Quake III Howto.
Linux Questions' 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 the official minimum requirements. It's a beautiful game which should really be played with the graphical settings set to "medium" or "high". For this reason the author recommends against using entry level cards such as the FX5200. Celeron owners (cough) should have at least a 2400MHz chip.
The Official Linux Faq
Linux Questions Quake IV bug forums
Quake IV demo