There are tons of 1D games. Somebody else mentioned Mancala, and I'd also mention the venerable Game of Goose, which can become anything from Candyland to sophisticated things like Kramer and Kiesling's That's Life or Parlett's Hare & Tortoise. Hell, Monopoly is also 1D if we're willing to allow circuits like Mancala.

Good observation. Considering stacking of pieces maybe 1.5D though.

Backgammon, the game everyone's seen and at the same time nobody knows how to play :P

Mancala is roughly 1D too!