Hi Turbocharged (and welcome to these forums).
I have a Canon 7D, and I have the same experience as you state with your new 5DmkIII (That is, after zooming in to review a photo, the quick control wheel on the back of the camera body does not move to the next photo).
However this is NOT a 'problem' I believe, but a built in function of the camera/s. So when one zooms in, the joystick can be used to move the 'zoomed in' part around, but the quick control wheel will only move to the next photo when you are viewing the full photo (or from memory, a page of 'multiple photos' (eg 4 photo thumbnails, or 9 photo thumbnails).
When you say that it doesn't happen every time, I think you need to 'zoom back out' to 100% (rather '1 full image displayed on the screen') to progress to the next one. That's my thought....
I hope this helps.