I'm experiencing this same issue, with the final release Linux Mint 13 "Maya" MATE.
I just installed Maya over the weekend. When I try to use 'Menu/Logout/Switch User' or 'Menu/Administration/Login Window' I receive a dialog that says "MDM (GNOME Display Manager) is not running." The dialog goes on to say that I might be using some other display manager, shouldn't be the case.
It is oddly intermittent, and I have tried installing LightDM, then tried uninstalling then reinstalling MDM and, as werkman2 reported, it seems to help for a while but the problem recurrs soon after.
My system is rather old - built in late 2000: MSI K7T266Pro2-RU (MS6380), AMD Athlon XP 1500+ (1350Mhz), 1GB RAM, 2X 40GB HDD, NVidia GeForce 2 MX400. Clean install Mint 13 MATE, linux-3.2.0-23-generic-pae i686.
MSI K7T266Pro2-RU (MS6380), AMD Athlon XP1500+, 1GB RAM, 2x 40GB HDDs, NVidia GeForce 2 MX400. Mint 9 Isadora (Gnome2) and Mint 13 Maya (XFCE 4.10)