Keyboard shortcut to move windows between monitors?

As @jrg mentioned, this is still possible as Unity (Unity 3D, not the low-power 2D version) is largely built using Compiz.

You'll need to install the Compiz Config Settings Manager Yourself though. Precise steps are:-

Install CompizConfig Settings Manager using apt-get install compizconfig-settings-manager compiz-plugins-extra.
Run it (just type Compiz in the dash and you'll see it).
Click Window Management.
Enable the Put plug-in (select the check-box).
Click on Put.
Configure the shortcut for Put to next Output (click the pencil icon next to the keyboard shortcut for Put to next Output.
Logout and login again for the changes to take effect (from the comments!)

This is based on my experience without the need for Compiz or any such utility.

When a window is activated (i.e. in-focus) If you press Alt + Space the top menu appears which has options like:

Minimize
Maximize
Move
Resize
etc.

Once that menu appears you need to press M key on your keyboard to select the Move option

Then you can just use the arrow keys to move the windows anywhere (even across monitors)

Compiz Config Settings Manager's Put plug-in works great on 13.04 and 14.04

Special hint:

Click on Put -> Misc. Options -> Activate "Avoid Offscreen": Avoids putting parts of the moved window to a hidden area. Very useful if someone has got monitors with different display resolutions.

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