Connections from Linux has always been available, but I just realized that connections to Linux computers are now supported! Not sure how I missed this: apparently this beta download was first announced on 2014-07-07. For Ubuntu and other Debian-based systems, there are both 32-bit and 64-bit versions of the application available.
All you have to do is install the Google Chrome Remote extension on both computers and follow the instructions to enable remote access. Press 'connect', type in a PIN, and you're in!
In Ubuntu, you can even customize what window manager runs when Chrome Remote connects by customizing a script at ~/.chrome-remote-desktop-session (remember to make it executable with chmod ugo+x ~/.chrome-remote-desktop-session).
If all you want is terminal access:
#!/bin/bash
exec xterm
For a nice lightweight graphical interface, try IceWM (install it first with sudo apt-get install icewm):
#!/bin/bash
exec /usr/bin/icewm-session
For Ubuntu's default Unity interface, try something like this:
#!/bin/bash
export DESKTOP_SESSION=ubuntu
export XDG_CURRENT_DESKTOP=Unity
exec /usr/sbin/lightdm-session 'gnome-session --session=ubuntu'
To make the new configuration stick, run:
sudo /etc/init.d/chrome-remote-desktop stop
sudo /etc/init.d/chrome-remote-desktop start