Also known as side-scrollers, or run-n-jump games, there are more of these for Linux than I had expected. Here are nearly all the ones available through the repositories. Most seem to have a version for Windows (indicated by the “W” in parentheses).
Alex the Alligator – a deliberately retro style, this has Gameboy graphics and sound, and seems fairly simple. Actually the fourth game in series. Has an edit mode that allows you to create new levels. It looks like only 2 levels are in the basic game.
Abe’s amazing Adventure – a full screen only game, decent graphics, you have to dodge some enemies and pick up items (including a gun). There are doors you need to find keys for. This game hasn’t been updated since 2003.
Asylum – This game starts off in windowed mode, has nice music, decent graphics, adjustable size window. Can be played full screen.
Blob Wars: Metal Blob Solid – Doesn’t look bad, although the graphics are a bit small. Collect items, rescue friends, kill enemies.
Frogatto – Created by the team who brought us the strategy game Battle for Wesnoth, this has excellent graphics. Lots of dialogue, including helpful hints on controls and playing. Can be run windowed or fullscreen.
Holotz’s Castle – Features ominous music, 2 characters to control, and seems to be a timed key collection game. The enemies can do one-shot kills, and there is dialogue (and I guess a plot).
Neverball – this one is insanely difficult. Lots of levels, more than one way to score points, but your mouse directly controls the tilting of the play field. Good luck!
Nikwi Deluxe – In this one, you are a kid who needs to collect all the candy items in each screen. Very colorful graphics. There aren’t any ways to kill the enemies (in the early game anyway), so you’ll need to dodge them while collecting all the items. Beware, some bugs can hit you from above with “droplets”. Apparently there are 30 levels, with a password for each. It can be run fullscreen if you want, by just running from console: “Nikwi –fullscreen”
Pachi el Marciano – An original art style sets this one apart. You can play windowed or fullscreen. There is “help” that gives instructions, and several difficulty levels. Needs to be run from the console. Also features a nice intro screen.
Secret Maryo Chronicles – This game is frequently mentioned when discussing “best games for Linux”. It has colorful graphics, a timer for best speed, a level editor, and gives instructions or hints on certain blocks.
Supertux 2 – This one, with SMC, always is featured on lists of “best Linux games”, for good reason. Not an exceptionally difficult game, the graphics are some of the best you’ll see. Features an introductory scene, sound effects. You are timed, but only for scoring – you don’t have a countdown like Super Mario Brothers has.
Teeworlds – This is an interesting mix, a platformer that is multiplayer only, AND a shooter as well. Pick up a gun, run around and find your opponents. (No screenshot because the game captures your mouse. Didn’t quite figure out how to escape it. Also, the mouse feels a bit sluggish)
Nebulus/Tower Toppler – This is a clone or port of a classic game that has been on many early platforms – the Commodore 64, the NES console, and others. Also known as Castleian, it features music and sound effects. I found it rather frustrating because not only are you timed, but you have to essentially memorize the path to take and when enemies will need to be dodged.