You're quite right about the mites, which can colonise the hollow stems of the straw, so it's best avoided. And of course mites can and will get into the nestboxes as well as on the floor of the coop, in fact this is more likely as they will be in contact with the birds when they go in to lay. shavings are very nice in the nestboxes, soft and absorbent and comfortable, so you could just use those. I don't use hay, it looks nice and comfy but soemetimes it can cause repiratory diseases in chickens because of the dust and spores - a bit like hayfever.
What sort of coop have you got? Some people do put shavings or other bedding on the floor, but if the chickens sleep on the perches it doesn't do anything to keep them warm, nor do they need it for this purpose. It's OK to just put newspaper down on the floor and then just roll up the top layer or two each morning into a sort of poo parcel and add it to the compost heap. Whatever you use, it's good to clean it out daily, not only to keep it fresh in there, but also to remove the overnight dampness in the bedding or floor covering, along with the solid droppings.
