One and Thirty is played the same as Twenty-One except (I bet you saw this coming) the total hand should come to thirty-one instead of twenty-one.
Another method of play is found in Early Card Games. Here, three cards are dealt to each player beginning at the dealers left. Play goes around the table in Rummy style, with each player picking up a card from either the discard pile or the undealt pile. When a player has a hand he wishes to play, he raps twice on the table. The remaining players have one more turn before all show their cards. The closest to 31 wins the pot. This book indicates also that somebody with three-of-a-kind has 30 and 1/2 points for that hand regardless of the point total of the cards, that an Ace is automatically 11 points, and that getting an exact 31 is an automatic win with needing to rap.
Bone-Ace
This same reference gives another version of a 31-style game. This is called bone ace. Three cards are dealt as before, but one card is dealt face up. The person with the highest face-up card receives one stake from each player. In case of a tie, the first person dealt their card receives the bets. The Ace of Diamonds, the "Bone-Ace", beats all other cards. The remaining game is, of course, who has the closest to 31. There is no mention of betting in this source, but I can see how the game could continue with betting similar to poker. However, no cards are to be exchanged.