My understanding is that Compiz does not work with Cinnamon but will work
with Mate. This is because Cinnamon is GNOME 3 and it uses Mutter as its
window manager. Mate on the other hand is GNOME 2 and it should work as
well as the old GNOME 2 desktop. You need to issue the command compiz
--replace. Google that if you are unsure how to do that.

I have read about Cinnamon working in Ubuntu with Compiz.
askubuntu.com/.../compizconfig-settings-manager-enhanced-zoo\
m-desktop-feature-doesnt-work-with

Configuring Compiz once installed is a matter of installing CCSM.12.04
(CompizConfig Settings Manager). Compiz is pretty much under Canonical
these days. They pay the developer and it is made to work in Ubuntu. GNOME
went with Mutter as it s window manager and it does not have all of the
desktop effects that Compiz has (by a long shot). Basically GNOME does not
want users chnaging things up. Apparently Mint is working on improving
Mutter's desktop effects.

Also so graphics cards may not work as well when you start making changes
due to driver issues.

Let us know how it turns out for you. I am not a Mate or Cinnamon user. I
have them installed, but did not like them, so cannot give more help than
this. Also I am running on a Kubuntu 12.10 base and not Mint 13, which is
based on 12.04. Mint 14 should be along soon. It usually lags Ubuntu by a
month or more.