Can I overlay lumber perpendicular to floor joists to raise the floor height?
I would say that the main issue with putting down boards like that is cupping or bowing of the wood especially since this is a high moisture area; that's dirt beneath the joists, right? Look at any wooden deck and you will quickly see what I mean by bowing or cupping.
Additionally if you haphazardly layout the boards like that then there will obviously be unsupported areas which will sag as soon as someone steps on it.
I recommend furring up the floor joists by using a table saw to rip 2 inch pieces of wood and attach them directly to the top of the joist.
If you don't have a table saw then you should be able to buy furring strips directly from a lumber yard or supply store.