Most clinics do. As for the doctors I think that would be a case by case things. We always pay cash- most specialists charge 1000-2000 for an appointment and if tests are required and you have to see them to review the tests they don't recharge you, at least that has been our experience.
For an example this clinic listed below is in Punta Cana. It is very large, very modern, and has many specialists. When you go for treatment, xRays, have an appointment, or make one for immediate treatment by a Doctor you first need to "check in". When checking in if you have insurance you present your documents. If you don't you tell them. Now most DR medical insurance covers 80% of your visit. So you pay the rest either by cash or card at this desk. So here the Clinic takes the money for the Doctors. Other places where you go straight to the Doctor they may or may not take Cards. Most do not, but more are taking cards lately.