It looks like currently you can't just plug it in and have Shotwell work out-of-the-box, but if you don't mind a little work, you can mount the camera via "PTP" mode and then import the images from the filesystem. See this blog entry from someone who did this on FreeBSD; the Linux situation will be similar.

Probably eventually the software will gain direct support, but honestly I wouldn't bother. I find using a card reader and swapping cards to be a easier workflow — I always make sure I have a fresh card in the camera, and I swap out the battery at the same time. Any card reader should just work in Linux, so you should be all set.

That's if you just want to copy JPEG files. If you want RAW, the situation is worse. The Fujifilm X-E1, like the X-Pro, uses a unique sensor layout and while dcraw, the basic RAW file decoder for Linux can read these files, I'm not aware of any RAW converter on Linux which will process them.

So, under Linux, you're stuck with JPEG only; you can convert RAW files to JPEG after the shot in-camera (although Fujifilm's implementation of that is clunky and limited), but out of the camera you're really out of luck. On the plus side, at least, the JPEG engine is amazing and you should be able to get great out-of-camera results. You just won't be able to count on the complete post-processing flexiblity of RAW.