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There is a noise issue, even with files at low ISO's. There not horrible, but they are there, and yes its probably only pixel peeping that you can see them.
Its sort of like a small scratch on your car, everyone else goes what a great finish, yet you know where that scratch is and it bugs you everytime you see it.
But the point is that, if you up scale 1D Mark III images to match the resolution of the 7D, the noise of the up scaled 1D Mark III files will be very similar to those from the 7D, and that will be true at the pixel level, too. So, if you are are bothered by noise that isn't a problem at the image level but only at the pixel level, the 1D Mark III files should bother you to an extent similar to the level at which the 7D files bother you.
Among the many DSLRs that I've owned, there have been a 1D Mark III and 7D. I essentially process(ed) files from both cameras the same way, and the same thing is true of my 1D Mark IV files. On several occasions, both on line and in person with prints, I've challenged those who say that they don't like the 7D image quality to determine, from a group of 8 pictures, which were taken with the 7D and which were taken with the 1D Mark IV. Out of the dozens of people who have taken a shot at that challenge, I'm yet to see a single instance where someone could tell me which images were taken with which camera at a level of accuracy any greater than what would be expected with random guesses. And I'm confident that I could do the same thing with a group of 7D and 1D Mark III images, and even though those who say that they can immediately recognize the allegedly poor quality of 7D images would not be able to accurately identify which images were produced by which camera. And all of this goes along with the strong belief that I've developed that, even if there are real image quality differences between different recently produced cameras (or lenses), the extent of those differences is not nearly as large as one would be lead to believe from reading posts on internet forums.