How Can I Reduce Noise On a 5D Mk3?
Certainly using RAW on ML will help.  With 14 bit color, your noise reduction options are a lot more advanced.  Traditional noise reduction that you are used to from photos can be used with video as well using either color grading software or editing software.  (With RAW video, it would be in the color grading software before mixing down to processed clips.)  Additionally, with RAW, you won't have the encoding artifact which you may be misinterpreting as noise as well.

That said, how bad of noise are you getting.  I'm in the reverse situation of you.  I started as a video guy and switched to photography and got the 5D Mark iii because of it's strengths in both areas.  You should not be getting significant noise issues below 800 ISO on your 5D Mark iii unless something is wrong.  The 5D Mark iii shoots outstanding video and with RAW is compared fairly closely (though still noticeably different from) much more expensive cameras like the C300 or even some lower end ARRI cameras.

It is generally applauded for it's high video quality and good low light video capability.  Can you post a sample that demonstrates the problem you are seeing?  I wonder if there might be some other issue causing your problems as my 5D Mark iii doesn't give what I would consider unreasonable noise levels on video even at ISOs like 3200 or even 6400 in some lighting conditions.

ISO 3200 video frame grab



There is some noise, but it's pretty limited, particularly considering this is in the shadows under a desk, inside, with no other light other than what was coming in through the window above the table.  There are some newer cameras in the same or lower price range that have slightly improved on the capability, but the 5D mark iii is still one of the top contenders last time I looked around.

Other than that, the easiest option, like Jason Conrad mentioned, is to use more light and lower numbered ISO.