If the grain is good, you can treat them like any board; no need to chase straight grained board of hickory/elm/oak/ash/maple.
If the grain is less than perfect, you could chase a ring, but you need to be careful there'll be enough meat left in the stave. If the 2x2 was cut at an angle, you may risk thin spots in the bow. Mark one ring along the enture length of the 2x2 and decide if there's enough wood left in thickness.
If the grain is less than perfect, you could also back the board (instead of chasing a ring). A backing of straight grained wood (or bamboo) will work, but rawhide or linen cloth are fine too.
What wood is it?
And how straight is the grain? It must not "appear" to have straight grain, but it simply must have straight grain.
