In this paper, we extend multiple results previously known for map folds on rectangular grids, to maps on an equilateral triangle grid. Specifically, we show that it is always possible to fold a 1D equilateral triangle strip in the general folding model, while it is (strongly) NP-complete to determine whether various subsets of an equilateral triangle grid can be folded via various simple folds; in particular for the assigned polyiamond paper in all models, and for assigned convex paper and unassigned polyiamond paper in the some- and all-layers models.