I hiked R2R2R with a few XC friends (2 girls, 1 other guy) last summer late July. We actually hiked most of it at night because my brothers had ran it around the same time and said it was HORRIBLE during the day....just plain miserable.
I must say, it was nice at night but we didn't get to see a lot of the canyon. We were only in the daylight about 4 hours for the entire trip there and back (minus during the day when we were at the N rim), and when we were hiking in the daylight, it was BEAUTIFUL.
I would highly recommend finding a more mild time of year (later in the fall or early spring) to hike it if possible so you can really enjoy the beauty, or at least avoid the midday heat if you do hike it in the summer.
We started hiking from S rim just as it was turning dark, got to N rim an hour after it got light, slept for a few hours, checked out the sights, then started hiking a few hours before it got dark again that night. Those first 3 hours hiking back were absoluely BEAUTIFUL!
Finally, I would recommend running it rather than hiking, as it goes by a lot faster, more engaging, you can enjoy yourself more when you get to N rim, and still get to see the whole route. It got a bit long at time just hiking.