How should I move a sprite in a direction based on the rotation of a child?

I want to move hexagons around on a grid in the direction of an arrow.
53935-arrows-directions.jpg

I set the rotation of the arrow gameobject (a child of the hexagon):

movementDirection = Vector3.zero;
movementDirection.z = Random.Range (0, 6) * 60;
arrow.transform.localEulerAngles = movementDirection;

How should I move the parent object in the direction the arrow is pointing?

I used transform.up which works perfectly.

Thanks to ActionHotDog on reddit.