Energy efficient multicast routing is one of the fundamental problems in wireless sensor networks (WSNs). Previous work has shown that when the goal is to find multicast trees with minimum transmission cost, the problem becomes NP-complete. In this work, we present a heuristic distributed minimum transmission multicast routing protocol (MTMRP) for WSNs. By introducing the biased backoff scheme and taking advantage of the broadcast nature of wireless communication, MTMRP chooses the forwarding routes which can connect more multicast receivers. Moreover, MTMRP introduces a path handover scheme, which can prune redundant routes for multicast routing. As a result, the multicast transmission cost is reduced in a distributed manner. We conduct extensive evaluations to study the performance of the proposed MTMRP compared with existing protocols. Simulation results demonstrate that our scheme effectively improves the multicast routing energy efficiency.