Pretty much any hotel will take cash for payment. They will probably want a cash deposit also or an imprint of your credit card in case you trash the place.
They are going to want to see your drivers lisence or some other form of ID to get the room though.
I used to have a mail box rented at a UPS store that I used for an address instead of my house. Cost 120.00 per year, all my mail went there instead of my house. That way there were no worries if the hotel decided to send you a thank you card or a discount voucher for future visits.
I've never tried it but a passport would probably work for an ID and it doesn't have your address on it.