This'll make you laugh.
Until a few weeks ago I was able to tether my D610 to Lightroom CC - now I can't.
At great expense I've spoken to Adobe Support in the U.S (I'm in the U.K) and guess what - they actually agreed that their software is currently broken. Specifically, Lightroom CC 2015.2.1 running on Mac OS 10.11 (El Capitan) has "many problems". Yes. Really. That's what he said.
So, the answer to the original question is that once upon a time it was not possible to tether a D610. Then it was possible. Now it isn't possible.
Brilliant. And they still keep taking my CC subscription.
If you want to know more detail, here goes...
Connect and switch on camera
File -> Tethered Capture -> Start Tethered Capture...
Enter Tethered Capture Settings (session name etc)
Tethering "control bar" appears
Note: "No camera Detected"
Really? Are you sure?
So why is it that I can click on Import and quite happily import/view images on the camera?