An illustrative example elevator sheave cleaner includes a base configured to be situated on a first side of load bearing members that wrap at least partially around a sheave. A plurality of fingers extend from the base and are configured to be received between adjacent load bearing members. A plurality of cleaning pads are respectively supported by the fingers and spaced from the base. The cleaning pads are configured to contact and clean a surface of the sheave when the cleaning pads are on a second, opposite side of the load bearing members.