I don't think tethered capture for that camera is supported. When Adobe announces that a new camera is supported, that generally just means support for the raw images from that camera. Tethered capture support is something different and there are only a few cameras that have it.
The Lightroom doesnt support still EOS 5d Mark III for tethering.
Maybe you could influence Canon to provide whatever help and documentation they need to provide Adobe with to complete the capture support. Canon I am sure would prefer you to use the software they provide for this purpose. Also this is a user to user forum and you should post your feature request on this site. see link.
My Canon 600D isn't supported yet in LR for tethering either. I tried a procedure at the below link using Canon's EOS Utility and LR's 'Auto Import' function. It works quite well once you have everything setup properly.
Scroll down to the section titled 'How to shoot with EOS Utility and LR' for details. The nice part about this is you have the option to review and set most of your camera's controls onscreen. You can also use the 'Live Preview' mode to compose and adjust the image real-time on your monitor. It would be nice if Adobe could incorporate live view and at least some camera controls for the Canon and Nikon DSLRs already supported for tethered capture. For now this procedure is a viable solution.
There are also options in the Canon EOS Utility Preferences for 'Save also on the camera's memory card' and 'Auto Power Off' disable to prevent losing your camera's USB connection.
While it may be true that Canon and Nikon are both notoriously slow on releasing SDK's so Adobe can add more cameras to the supported thethering list ... we also must acknowledge that shooting tethered in Lr has become much like playing the lottery ... some folks have great success ... for many others it can be an exercise in futility ... it isn't a very consistent feature ... passing all the blame to the camera makers can, and does get old ... Adobe either needs to directly address and correct these issues ... or consider dropping the feature from their marketing fare ...
Nikon released their SDK's for the D4 and D800 series quite some time ago and for the D800 looks like it was in 2011? If that is the case, going to the SDK site could be a good source for new and unannounced products
Canon do not seem to be as open with the SDK kits so maybe that is why LR4 supports the new Nikons and not the Canon?