We Are Dancing in the Forest (Wolf Game)

This game is very popular with my students and can either be a chase or freeze game.
Source:  Eisen, Ann and Robertson, Lamar.  An American Methodology: An Inclusive Approach to Musical Literacy p. 402

Up, Down

This game is great for the primary grades, especially at Halloween.
Source: Trinka, Jill.  The Little Black Bull and Other Folk Songs, Singing Games, and Play Parties. Folk Music Works. 1996. pp. 50-51

Snail, Snail

This is an English children's game in which a leader slowly leads a line into and out of a spiral shape. 
Source: Locke, Eleanor G.  Sail Away: 155 American Folk Songs to Sing, Read, and Play. p. 4

Plainsies Clapsies

This simple three-note English ball-bouncing song gets complicated and fun when you try to sing it while doing all the tricks.
Sources: Eisen, Ann and Robertson, Lamar.  An American Methodology: An Inclusive Approach to Musical Literacy. p. 374
Chosky, Lois.  The Kodaly Method.  Prentice-Hall, Englewood Cliffs, NJ. 1988. page 170.

Pizza Pizza

Upper elementary students giggle and cringe when they have to choose who their "boyfriend" or "girlfriend" is when they play this very active game, but still request it frequently.  I sometimes only ask them to name a favorite pizza topping.  This way is less controversial, but not as hilarious.
Source: Locke, Eleanor G. Sail Away: 155 American Folk Songs to Sing, Read, and Play p. 6
See Also: Pizza, Pizza Mighty Moe as written by Jones, Bessie and Hawes, Bess Lomax.  Step it Down: Games, Plays, Songs, and Stories from the Afro-American Heritage. p. 63