Parallel Perfection Origami Tessellation
I was messing around with more complicated folds and not really succeeding, when this idea just jumped out at me. So I went with it.
Collapsed hexagons in one direction and elongated rhombuses in the other. Both the rhombuses and the hexes are natural to the triangle grid.
Some offset triangles result in the spaces between.
It's a pretty simple design. Fairly easy to fold. Any old paper will do.
I have included a basic crease pattern for those that want it.