I remember few of the games we played during recess. If we were outdoors I remember playing dodge ball, a game that is outlawed in most schools today (too violent), but which I found to be a lot of fun. Dodging that ball built my "cat like reflexes" (just kidding) by making me learn how to run backwards, forwards, sideways, ducking under, jumping over, etc.
Another game we played was Red Rover -- Red Rover, Red Rover for pink to come over. If you were "it" you had to guess what colors were active and who had what color. If you guessed a usable color, your turn ended, and the person who had the called color was it. Rather boring I always thought, but some classmates liked it.
My all time favorite game was an indoor game. It was Hucklebuckle Beanstalk. The teacher started it off by telling us all to hide our eyes and not cheat. We wouldn't dream of doing that in Kindergarten. She would then hide an eraser (the paper/pencil kind) and the "it" person had to find it. I always wanted to be "it". Then I would get to find the eraser and subsequently hide it.
That game depended on your classmates accurately telling you whether you were hot (near the prize) or cold (far away from the prize) correctly. I played this with my own children and their playmates on indoor days.