You're doing the right thing, finding places where you can promote the game. In the future, you might want to just post a thread asking if you can promote your Linux game in a single promotional thread, explain your affiliation, and maybe offer to give out a few free licenses to the forum members.
I have a website that I've grown from a tiny thing to a site that gets tens of thousands of visitors. (It's https://linux-tips.us
if you're curious.) You can also try approaching various Linux blogs/educational sites, asking if you can (again, be perfectly honest) write a promotional article and have them post it.
Again, in those instances, you could probably offer a few free keys to those who want to try it.
There are a ton of ways to promote your site. I'm completely new to SEO (and certainly at this level) and mostly got the viewers I get due to smashing out a bunch of articles - one every other day without fail for going on two years.
If you're creative, you can promote the heck out of it. Feel free to start an off-topic thread asking about how folks would promote your game. You never know - and it won't hurt to ask.
I can relate. I put a whole lot of hours into helping the Linux community, including my site. I monitor that bad boy all the time and make continued improvements. It's a labor of love 'cause I barely get ad revenue and have to pay for the hosting and the CDN bandwidth.
You can always do your own community. Install a forum, add a blog to remark on new updates and game changes, and just keep at it - hoping it develops a community around the game. Maybe have some sort of tournaments where the winner(s) get their name posted on a "Who's Who" page. It doesn't even need to be cash prizes or the likes.
Then, be patient AND keep up the work.
You can also promote the site via social media, like Reddit. It's a Linux game, so make a post on Slashdot. It's a Linux game, make a post on Hacker News. Better yet, get one of your users to do it for you. (I don't have time or motive to promote your game for you. Someone else might. You just need some fans.)
I don't really want to derail your thread, but SEO is important.
If you use obscure keywords - like "Linux chess game" and pay for ads with AdSense, you might get some quality traffic that doesn't cost a whole lot. (Yeah, that's paying for promotion. You can do a lot for free.
I don't want to derail this thread, but you have options. You have all sorts of options to promote your site.
Wait a week and approach another Linux forum asking if you can promote your game there. Just disclose it, write it up showing how you're affiliated, and ask in a thread ahead of time. I really want to allow this thread through the spam queue, but I really couldn't do so until you did those things. Otherwise, it'd have been considered spam and I figured you had the best of intentions.