I don't know if Pineapple and Cursewell qualify as "off-beat" poker games, but they are popular games among our friends anyway. And lots of fun too
Pineapple (or Crazy Pineapple) is quite well known I guess. Here are the rules:
Each player is dealt three holecards (instead of two) at the start. One of these cards is discarded before the flop (in regular Pineapple) or after the flop[after betting] (in Crazy Pineapple, which is the more common version). The betting is the same as Holdem. At the showdown, a player can use any combination of his remaining two cards and the boardcards. A player who still has three cards in his hand at the showdown has a dead hand.
Cursewell is five card Omaha game. Rules are same as in Omaha with two differences.
1. Each player is dealt five holecards.
2. The first community card is dealt before the first betting round, so that each player has five private cards and the single community card on his first bet. Then two more community cards are dealt, and play proceeds exactly as in Omaha.
Cursewell is a variant of French poker game named Courcheval. Courcheval is played with four hole cards like regular Omaha.