Here in Iowa low till and no till practices have been the norm for the last twenty years. Anything that isn't being sold to the elevator is left on the field and gently tilled in for the next year.

Wow! Good. Here in Oregon they take the wheat and then the straw for animal bedding. Same with grass seed(big crop for us here)

Here they grow corn and soy beans in rotating. Wheat plantings are small and always grown to get the straw - the wheat itself is considered a by product. I'm sure its done differently in other parts of the world ... straw is a crop, too.