Quite a few action style games available in the Gentoo ebuild repository.
Chromium B.S.U. is a classic arcade raster graphics vertical scrolling shoot 'em up single player game. It was written with games like Raptor in mind and the idea to enhance them for modern graphics. As such it uses opengl via media-libs/freeglut or alternatively sdl, supports fullscreen and modern audio themes. View a YouTube demo here.
emerge --ask games-action/chromium-bsu
Minetest is an infinite-world block sandbox game and engine, heavily inspired by Minecraft and InfiniMiner. The majority of ingame content is provided by the community in the form of mods, as well as texture packs. Mods are installed on the server-side and there are already quite a few dedicated servers running you can join. The engine is written in C/C++ and is meant to be portable and lightweight and should run even on fairly old hardware. Supported platforms are Linux, OS X, FreeBSD and others. Features include building, crafting, multiplayer, lightning graphics and a map generator. View a YouTube demo here.
emerge --ask games-action/minetest
Supertuxkart is a free kart racing game starring Tux. It's a fork of TuxKart and under active development. It features single-player, multi-player and a story mode. You can choose from a lot of funny characters including Tux, Mozilla, Pidgin, Puffy and of course the GNU. There is also an addon manager integrated, you can find some addons here. The gameplay combines regular kart racing with the possibility to collect items on the tracks with different properties (such as a cake bomb or a glue-like stuff to slow down opponents). You also have to look out for fuel items you can use to boost your speed for a short period of time. This makes the gameplay much more fun and reminds a bit of Super Mario Kart. View a Vimeo demo here.
emerge --ask games-action/supertuxkart