Not necessarily, its all personal preference at the end of the day some may like some may not. If you really want to go with the joist then put a spruce ply on top and refasten the osb. You may see waves in your floor if you just go parallel over the osb. Crossing the joist makes the floor stronger and hides dips/humps between joists. If you beef up your subfloor you wont have to worry about crossing the joists.
I would definitely refasten the subfloor regardless before installation.