How can I use a CyanogenMod theme with a stock rooted Galaxy S III?
You cannot. The Cyanogen theming engine is specific to CM and derivative ROMs.

Samsung stock ROMs have Touchwiz UI, which is a proprietary layer. Theming CM to look like Touchwiz, or creating mashups would be easier because of CM's open nature. Some carriers might give packs, like Sprint (source).

You can install a different launcher and install themes on it, of course. Go Launcher, for one, has  many themes.