You can go to the main landfill in Johnston. It's called the Rhode Island Resource Recovery Corporation.

If what you're tossing is general demolition debris, there's a special drop house for C&D (construction & demolition). I've brought pickup truck loads there before.

Vinagro up the road also has a C&D yard but I'm not sure if they bill small customers. I've only ever seen from the junk haulers on to huge walking floor trailers there. My company has an account with them, so everything was done by waybills when I've dropped there.