This is a game I played with my little host sister. I actually thought it was あっち見てポイ, but doing some searching I see that the 見て (look) is probably 向いて (turn; face), though I suppose in varying regions it may be different. Also, some people say ホイ instead of ポイ.
Here's how to play:
1) Play Rock, Paper, Scissors.
This is called Janken ジャンケン in Japanese. How you play the game (at what speed, with what hand movements, with what words) differs depending on where you are, so I would recommend just playing by your own rules.
The way that I learned to play in Japan is, you say 最初はグー、ジャンケンポイ！(First is rock, janken poi!), setting your hand at the ポイ. The rhythm of your hand movement seemed to be to down-stroke on every other syllable (which took me a while to get used to, since at my house we move our hand at every word and set our hand at scissors instead of 'shoot'). There is probably a different way of playing rock paper scissors for every 100 square miles of land on Earth.
Note: In the Japanese version I learned, if you tie you try to break the tie by saying あいこでしょ, setting your hand at でしょ. I'm not sure what aiko desho means; sometimes I hear it is used just to keep time. If you tie again, you keep on with the あいこでしょuntil a winner emerges.
2) Have you won?
If you've won the janken, you get to move on to the next half of the game. Point directly at your opponent's face. Your goal is to point in one of the four directions and have your opponent look in that direction, so try to keep your decision hidden (no shifty eyes!). Say あっち向いてポイin a measured way (to keep time), and at the ポイ you must point your finger in the chosen direction. My brothers don't speak Japanese and couldn't remember the phrase, so we just say "Look over there", 'there' functioning like ポイ. At the same moment, your opponent (whose goal is to NOT look in the direction you point) will move his head to face in their chosen direction. If the direction you point and he looks is the same, you have won. If not, go back to step one.
3) Have you won again?
If you've won the あっち向いてポイ part, you get to bask in your own glory a moment before starting the next game; or you get to deal out a punishment to the loser. My host sister and I played without any ばつ of course, but if you want to add some consequences to your game, it only makes it more exciting. At Nanzan the common punishment was a flick to the forehead with the middle finger. This is the worst punishment of all. If you are the loser, the winner will put his hand over your face, pry back his middle finger like a catapult, and then release it to wreak havoc on your third eye. If this is too extreme, I recommend the good ol' two finger slap to the wrist, or maybe an indian rug burn. If you teach this version of the game to boys, you will have them occupied on long car rides the entire trip, no matter how far the destination is.
Here's a video of a J-Pop group doing some あっち向いてポイ. I chose this one because when they're determining who is going to play who, they do something called グーとパーで別れ (separate with rock or paper). What you do is, everyone stands in a circle and at the last syllable of 別れ you choose either rock or paper. Until the group is evenly split by rock and paper, you keep doing this. It's an easy way to make teams, similar to the 'odd and even' or 'dickey dickey donkey' type of counting games that we do in America when splitting up to play tag, etc.