I used cash or debit cards successfully at Enterprise (had a great relationship with them) and also Thrifty. Was a no-go with Avis but was successful with Budget; had to arm twist them.
Per hotels, no bad problems at all. Secured with a debit during reservations and used the debit card or cash. Stayed in major chains/franchises like Sheraton, Radisson, Marriott, Residence Inn, Courtyard by Marriott, Hyatt, Embassy Suites, and Extended Stay. Did have a problem at Residence Inn - they insisted on charging me extra to pay with cash and to use as a deposit. That deposit was higher than what was advertised and higher than if I used a CC. Sheraton was very insistant that I use the same card that I used at reservation. I wanted to pay cash. However, I stayed in the same hotel many weekends before. Manager recognized me and profusely apologized. Stayed there many since without any problems.
