Thanks for all the feedback.
Yochanan- I'm sure it's just the lens, as every other lens I've tested on the Mark III is superbly sharp. My Sigma 70 2.8 macro is the sharpest lens I own (one of the sharpest lens anywhere) and it is incredible on the new camera, so I have no issues with the body.
I think I have figured out that my 24-105 has a resolution characteristic very similar to the tests depicted on the SLR Gear website. http://www.slrgear.com/reviews/showproduct.php/product/145/cat/11
If you look at their 3D blur plots, both on the crop sensor and the full frame sensor tests, it shows the lens to have it's worst performance at the 70 and 105 end, which is where I was comparing it with my 70-300L lens. It looks much better at 24, 35 and 50 on those plots. The edges at the higher focal lengths look pretty poor on their plots, but the centers are OK. I think this is the only lens I've seen on their plots where the best image is at f11 (on their 70mm test). Usually the sharpest margins are at 5.6 or 8.
I will do some more tests of the lens at the lower focal lengths on my T2i to compare it to my 15-85, but I suspect there won't be as much difference at the wide end as there is at the tele end. I do at some point need to consider getting a prime or two at the wide end, but if the 24-105 is acceptable at those focal lengths, I guess there is no hurry. May start with the cheapie 50 1.8 before I take the plunge into a 35L or a Zeiss.
I tend to shoot two types of things, landscape/scenery, where I'd like better full frame sharpness than I'm seeing at the tele end, and I also do a lot of flower photography, where the margins are usually not as important. The first day or two I had the new camera, that is what I was shooting mostly around the house and was fairly pleased with the results for flowers, even at the tele end of the range. It wasn't until I did the tests a few days later I realized the problems away from the center. For the time being I've decided to live with the lens, and do some more tests next weekend before deciding if I want to send it to Canon for calibration.