I've been tethering fine here with the same camera. Can you give more detail to trouble shoot? Is EOS Utility turned off? Is the USB port loose on the camera? Have you tried a different cable?
Which "Mark III"?
Thanks for replying. The cord/USB connection are both fine, as I can connect the camera (Canon 5D Mark III) to other applications with no problem. What seemed to have fixed it for the moment was to re-download and re-install the most recent firmware update (Jan. 2015) to the camera, we'll see how long it sticks. I tried everything else I could think of. I downloaded Lightroom 5 as a backup, because I was previously tethering to that with no problem.
I'm having this same issue. The EOS utility will allow me to shoot tethered but not Lightroom, it blinks from showing my Mark III to no camera detected. I'm using the latest version of LR CC version 2015.3. Any ideas on how to fix this?