Rib-eye is taken from a slab of meat just before the tenderloin (I believe the famous Delmonico Steak is the third rib-eye back from the chuck). I prefer mine with the bone. The meat slab has a grain that runs end to end, and the steaks are cut against the grain. If you wanted to further slice your steak against the grain, you would have to lay it on its side and slice the width of the steak, making a thinner steak. Like many others here, I prefer to just slap mine on the Webber, over a solid bed of charcoal, and let it cook until rare for me, medium rare of the DW. Season simply, with salt, or S&P. Serve with something refreshing, like a wet slaw, or a great melon salad. To me, the rib-eye is the best cut of meat on the carcass, period.
If you are tired of the same old, same old, because you have too many rib eyes, too frequently, try dicing the meat, and stir fry it with come fresh veggies, such as bias sliced carrot, onion, bok choy, snow peas, with a toouch of ginger and Kikoman Lite soy sauce. This is very tasty. Of course if you live in that world, you have far more disposable cash than me.
Seeeeeeeya; Goodweed of the North