Is there a keyboard shortcut to move a window from one monitor to another? Is there a free app that I can use to move a window from one monitor to the other monitor (assuming only two monitors) via the keyboard? I shouldn't have to use the mouse at all (e.g., for selecting the window or the monitor, or for dragging).

I know this can be done in Linux using compiz; I'm surprised there isn't a (free) option for OS X.
UPDATE: Spectacle is no longer maintaned.  The website forwards users to Rectangle, see Jack's answer or https://github.com/rxhanson/Rectangle/releases.

I just tried Spectacle, great app and it's free.
This is what it can do with the selected group of windows.

You need to activate accessibility support.
System Preferences &gt; Accessibility &gt; Enable access for assisted devices
As per comments: The accessibility option in Mavericks is now in
System Preferences &gt; Security and Privacy &gt; Privacy &gt; Accessibility