Coming later to this thread, I decided to add a reply here because there is so much good information in the older messages. I hope everyone read all the posts. Well, I should say I am glad I read all the posts because there are good suggestions to try when facing [apparently]slow auto focus on their Mk3's.
...because I am one of them.
I have had some good conversations with Canon. I am a professional photographer who will use the body for paid work when back home from this long personal travel, so I have to sort this out.
I have tried the camera with the lenses I have with me: 85mm f1.8, 24Lf1.4, and the 24-70mm The 24-70 has some kind of problem with focus motor and is going to Canon this morning. But even with the others there were delays. I KNOW this isn't the result of careful testing, just a description of my experience in the moment. I am adding a tile to the mosaic of shared experiences with the 5DMk3. I have been shooting with a 5DMk2 for a while. That body is at home.
I have spent several nights shooting with the 24 and 85 in a dark bar where local musicians get together to jam. I am sometimes pretty close. Not a concert -- the lighting is way lower than stage illumination. Some fluorescent light, too. I plan to use this body at wedding receptions, for example, and the challenge will be similar. I too had delays getting any confirmation of focus, and I was set up to require confirmation before shooting. Perhaps I will try allowing the camera to shoot before confirmation but it seem like a bad idea. I tried switching back to manual focus and it seemed more fluid and quick to get shots! Yikes! I was also practicing constant movement of the chosen focus points to train my without-looking muscle-memory-touch.
So I will keep testing, but because I am 6,000 miles closer to Canon's Virginia repair site I may make them swap out my body. Maybe the new one will do all the same things and I will know what I have to do to adjust. But the OP had a similar experience to mine and although there are some excellent suggestions in the thread (which I just stumbled on this morning) I have to be confident in the body.
Thank, especially, to all those people who contributed their thoughts, suggestions, detailed experience with AF settings, and personal testing.