Android 2.x is easy: plug 'n' play with linux.
But in Android 4.x that component was deprecated/removed, and you have to use "MTP" to connect. And most (all?) Linux distros have no support for "MTP".
Aside
Since writing this, what I actually do is let the device be detected as a camera. Then I open the "Pictures" folder, and just copy everything in there. Movies, PDFs, etc. it does not seem to matter. So I've given up being able to organize stuff intelligently for the convenience of that having to mess around with the below steps.
The Solution
1. Download go-mtpfs.x86_64 from http://hanwen.home.xs4all.nl/public/software/go-mtpfs/
2. Make it executable ( chmod 777 go-mtpfs.x86_64 )
3. Make a mount point directory. E.g. mkdir nexus7 (may have to be root depending on permissions of the parent directory)
4. Plugin Android 4.x device. Make sure it is using MTP.
5. Mount: sudo ./go-mtpfs.x86_64 nexus7
6. View: sudo nautilus nexus7 (you could also use the commandline for your file actions: sudo bash )
7. To stop I did three things: close nautilus, then press ctrl-c to stop the go-mtpfs program, then sudo umount nexus7
Warnings
That is using a pre-compiled binary, that you are going to run as root. Be afraid, be very afraid.
However it is an open source project and you can compile it yourself: https://github.com/hanwen/go-mtpfs/
You need to install the "go" language, and probably some other stuff.
I used the pre-compiled binary, the 28 Jun 2013 version; here is its md5sum:
$ md5sum go-mtpfs.x86_64
bdc71b5e92dbabfa6d33c03a8a7f0e09 go-mtpfs.x86_64
(This is meaningless without some trusted source telling us it is correct, as it may already have been hacked; but if you get a different value and it still says 28 Jun 2013, I'd be suspicious...)
Notes
This worked on Ubuntu 10.04, doing nothing extra than described above.
It should also work on Ubuntu 12.x
It is slow. 10MB file in 3.7 seconds seems typical. A 435MB and a 480MB file in 4m55s and 5m43s respectively (copied at the same time). A huge file is averaging 2.2MB/sec.
To make sure you are using MTP, go to settings, then Storage, then press the icon with three vertical dots, and choose "USB computer connection". The choices are MTP and PTP. PTP works out of the box with Linux, but only shows you the photographs directory. That is perfect for plugging in to a printer at your local 7-11 or Fujifilm shop (or local equivalent if you are not in Japan), as it does not expose any other files but your photos. But not useful for anything else.
I suspect you always have to run go-mtpfs as root, but there is probably a way to have the mount point be visible to normal users, not just root. For me it was easier to just run nautilus as root, than try to work out how. Comments welcome!
If you want an alternative, or just more background information, this page looks good: http://bernaerts.dyndns.org/linux/74-ubuntu/247-ubuntu-automount-nexus7-mtp
