We have a range of beautiful wooden Mancala/ Oware games from Africa. Best for schools would be the pencil starter set, which comes with authentic Bonduc seeds as the playing pieces.
Shogi is the Japanese version of Chess. The primary difference is that when pieces are captured they can be re-entered onto the board playing for the opposing side.
Crokinole is a Canadian game invented in the 1870s. Points are scored by flicking disks as near to the central hole as possible. The added twist that makes the game so engrossing is that your disk must move the disk last flicked by your opponent.
The moving tigers game (Bagha Chal) is the national game of Nepal. It is a game for two players, one has four tigers and the other twenty goats. The tigers try to eat the goats while the goats try to surround the tigers so that they cannot move.
Go, or Wei Chi, from Japan and China. Beautiful game that appears simple but is actually more complex than Chess.
Tawlbwrdd is the Welsh version of the old 9th century game of Tafl. A strategy game of unequal sides, the objective of the defenders is to get the King to the corner of the board unharmed while the attackers must try to capture the King.