I've had bikers go off on me for it, while literally pedalling a moped in a bike lane engine off. I tell them to fuck off.
Ohio Revised Code 4501.01(L) defines mopeds as being interchangeable with motorized bicycles. It's not a motor vehicle in Ohio it's a motor assisted bicycle. So bike lanes are okay. Seriously print this off and keep it with you. If you're doing more than 20 down a bike lane you may have issues.