Traditional - Madagascar

Fanorona is a 300-year-old game from Madagascar. There they play that while a piece is in a capturing sequence, it can't return to the same point twice. Although the board is formed by placing two Alquerque boards side by side, the methods of capture are entirely original.  [From Zillions' ZRF of Fanorona]

The board has the following shape:

where the stones are placed in the following way:

MOVE - Stones may move to one empty cell through the lines displayed on the board.
CAPTURE - A stone capture by approach: after it moves, an enemy stone (or an unbroken string of enemy stones) adjacent to it along its line of movement are taken. 
It may instead capture by withdrawal: if it begins its move adjacent to an enemy piece (or unbroken string) and moves directly away, the enemy stone(s) are taken.
After the first capture, the stone may continue to jump (not necessarily on the same direction) until there is no more captures to be done. it
may not immediately return to the point it just occupied.
A stone may stop any time after making one or more captures,
but at the start of the turn must play a capture if one exists.
GOAL - Wins the player that captures all enemy stones.

According to the book "Fanorona: The National Game of Madagascar" (by J. et S. Chauvicourt, 1972, translated from the French by Leonard Fox, 1984): Under the reign of Ralambo (1575-1610), Prince Andriantompokoindrindra, who was heir to the throne, became a famous player and his name has remained attached to the Fanorona engraved at Ambohimalaza. [...] Tantaran' ny Andriana" [1902], Father Callet states that Fanorona goes back to the Vazimba (the earliest inhabitants of Madagascar) and came to the island with the first settlers. The ancient people called it fandrao maty paika ('we are afraid of being blocked').

Fanorona can be play by email at Richard's web server. There is also a Java applet to play this game. There are two modern games that were inspired by this strange capture rule Bushka and Ultima.