Opensuse 12.2 here. With XFCE as the desktop it’s fine, with KDE4, QT seems to have a few problems.
I’ve gotten it to run successfully in JoliCloud. After I installed the ttf-mscorefonts package, it even opened my projects out of my Dropbox folder with the same fonts I was using in the Windows version.
Ubuntu 10.04, Openbox as WM.
Ubuntu 12.04 64-bit with Gnome
Mint 13 / MATE. No problems so far, but I just installed it.
I have just upgraded Ubuntu to 12.10 and everything works the same, at least in 32-bit land.
Ubuntu 12.04 32-bit with Unity.
Mint 13 and Cinnamon
I am using openSuSE 12.1 64-bit, w/KDE 4.7.
Had a few minor issues with some of the 32-bit libs not being present, but that was rather easy to resolve.
I use Kubuntu Linux, that is Ubuntu with the KDE desktop by default.
Using Linux Mint 14 with MATE (32-bit).
I had installed Ubuntu 12.10 first but was fighting Unity, so tried KDE. That made Scrivener not work. :-/ So I started over with Mint.
Wade.
Ubuntu 12.10 with Unity and ArchlInux with Gnome.
Ubuntu 12.10 and 12.04, works great with both Unity and Gnome… 32bit and 64bit.
I use Linux Mint 13 (Maya). Using Remastersys, I have created a live DVD based on Mint 13 that is packed full of free software for writers and publishers. Naturally, Scrivener is on there - David from L&L said I can I include the Linux beta. As well as that there is Celtx and Trelby for scriptwriters, Storybook 4 for novelists, Xmind (mind-mapping), SciTE, Amaya and Kompozer for web writers, GIMP and Inkscape for designing ebook covers etc, WriteType and Focuswriter (both simple uncluttered text editors and much nicer than the likes of Q10 and Darkroom), Sigil ( really good ePub editor ) etc…
Where they are freely available, PDF manuals are present too.
Since the advice most frequently given to writers is to sit down and write, I’ve removed distractions like games and email clients. I kept Firefox for research purposes.
The DVD is almost ready for public consumption and is free. If anyone feels like being a guinea pig and trying it out I’d be grateful for the feedback. You can download the iso here: https://docs.google.com/file/d/0B1p0AVqAVCtYU3l4cTRaMEw5MzA/edit?usp=sharing
Mark B
Debian Wheezy (and #! or aptosid on some of my systems).
Jesper
Moving to 64 bit Ubuntu 13-04 with Gnome 2(I think) from a weakening Mac Mini with dickey wifi running Mountain Lion.
Not tested much yet.
I use Debian (wheezy) and Free BSD as my two nix’s … but have not tried scrivener on neither, as of yet, since I just discovered the “good news”. I’m going to give installing a shot tomorrow. Thank you for developing a version for Tux! 
Elementary OS Luna. Everything OK, both install and work.
I used to be a die-hard Ubuntu fan, but given the current surveillance scandals and since Canonical sends all my searches to Amazon, I switched from Ubuntu to Debian.
However, I can’t get Scrivener to run on Debian. Bah. 
I had been using Scrivener on Chakra 64bit on both desktop and laptop. It worked okay, but I had no luck getting spellcheck to work. I finally broke down and added the 32bit version of Elementary OS Luna to the laptop and have been really pleased. It’s a great distro for a notebook - clean, fast and attractive…