It's delayed another week according to a Steam forum post by Justin from Double Fine:
"The Linux version will be released for anyone on Linux. It's delayed another week unfortunately, due to some tricky driver/GL bugs we still have to resolve. Sorry for the delay guys =/"
Original Post:Those of you who are waiting for the Linux version of The Cave won't have to wait very long.
Ron Gilbert has updated his blog with news that the Linux version of The Cave will be released next week.
Source: GozzoMan
EDIT: On the Steam forum, Justin confirms that The Cave doesn't support cloud across different OS, but you can manually copy savegame across them (I tried from Windows to Linux, it does seems to work). Here are the paths on the different platforms:
Win:
C:\Users\%USERNAME%\AppData\Roaming\Doublefine\TheCave\(numbers)
OSX:
User/Library/Application Support/Doublefine/TheCave/(numbers)
Linux:
$HOME/.local/share/Doublefine/TheCave/(numbers)
Just copy around the whole (numbers) directory.