I did the North Rim to South
I did the North Rim to South Rim in October. It was one of the best experiences of my life. I did it as a hike, not a run, mostly to enjoy the experience more. Plus, my knee was still sore from a marathon two weeks earlier. The Grand Canyon is not a place to enter with sore knees, but I still had a great time.
The weather was perfect. Not too cold on the North Rim, and not too hot at the bottom or on the hike up. In October, there is more shade on the South side.
I would strongly recommend doing a rim to rim, or a rim to river to rim, before trying the R2R2R. Also, try to go with someone that has done this before. The friend I went with has done the R2R six times and R2R2R once. He took care of all the details. I had done research before the hike, but didn't have a clue about things until actually hiking in the canyon.
I hope to do one more R2R in the spring, and hopefully complete a R2R2R in the fall.
If you do this, treat yourself to some good beer at the end. I don't mind PBR and I would rank it above Natural, Busch, Bud Light, and Miller.