The ribeye is my favorite cut and I have it as often as I can. (or sneak into the house) I have been fooling around with a reverse sear that is getting me some really great results.
I will rest the steak at least overnight uncovered in the fridge, with a generous salt and pepper application. When I am ready to prepare it I put it in a cold oven set for 170 degree for between 20-30 minutes- based on the thickness and who I am cooking it for, 20 minutes for a 1 inch thick, well marbled steak will give me a true Medium with just the texture change from rare that I like.
I then light my fire and get the lump and wood chunks Hickory/plum burning very hot. I run the steaks out and get them on the coals, shifting them once and flipping once.
This gets me a true "Pink on the inside and Crusty on the outside" with perfect x cross hatches.
There is a lot going on with this, from giving the salt time to work, to getting the surface of the meat dry for a good sear. I have really been enjoying these steaks cooked this way, and find the concept works really well with other products like sausage and even burgers.
I've also had a LOT of success cooking steaks using the "reverse" sear method. For steaks thicker than 1 inch, this is now my preferred method of cooking.
As Jim pointed out, salting well in advance of cooking is essential in my opinion. At least four hours before I want to cook a steak, I'll lightly salt all sides of the steak and let it sit in the fridge, uncovered or loosely covered with plastic wrap, flipping the steak over midway through the time so that one side doesn't sit in any excess liquid. However, you want to make sure you give the steak enough time to reabsorb the liquid that is initially drawn out by the salting. Overnight is usually safest to ensure that the liquid has been reabsorbed.
If I'm cooking outdoors, I'll put the steak on the grill (cold) as far away from the heat (I use charcoal dividers to push all the charcoal to one side), get the grill to go as low as possible, anywhere from 175 to 200 and let it slow roast. (For indoor cooking, depending on what else I'm cooking and when I want the steaks done, I'll cook it in the oven as Jim described on a rack over a sheet pan anywhere from 150 to 200 degrees.)
When it gets to 95 degrees inside, I remove the steak and let it rest, lift the grill, push the charcoal dividers together in the center, and open up all the vents and remove the lid to get the charcoal roaring hot. I re-season the steak with salt and pepper, and then sear it over direct heat on both sides until I get my desired temp of 130. The actual direct heat cooking time is usually less than 3 minutes a side.
I've noticed that the temperature rise after taking the meat off the grill is less than what is experienced when cooking the entire time over high heat. I assume that this is because less time is spent grilling over high heat. In my experience, I've noticed that the resting temperature rises about 5 degrees, rather than 10 degrees.