After a somewhat painful search (it's still very, very difficult for me to navigate around chinese web pages), I found some software that allows you to play 斗地主 (dou dizhu, aka the landlord game, aka the original variant of rich man, poor man/asshole/president/etc) against other people online in linux. Other than Yahoo Games Chinese (which doesn't work very well on my computer because all the fonts are wrong and the characters show up as rectangles), most of the online card/board game networks seem to require that you download their Windows-only game software, and I couldn't get any of them to work in wine.

To get it to work, I downloaded the full distribution, unpacked it, and ran it with:

cd BlueDJGames-3.1.2
LD_LIBRARY_PATH=.:Qt ./Hall

You also need to register an account before you can play. Once you get it running, it's localized both for Chinese and for English.

There's source code, but it seems to depend on some old version of QT.

There are other games in the distribution -- blackjack, a bunch of card games I don't know, mahjong, domino, chinese chess, go, something called Land Battle, and Shisen-sho, but I haven't tried any of them.