The amalgamation of 3D VLSI technology and Networks-on-Chip (NoCs) offers a promising architectural platform for the future Multi-Processor Systems-on-Chip (MPSoCs). Since multicast communication is frequently exploited in such systems, it is highly desirable to design NoC-based routing methods that support multicast. In this paper, a highly adaptive and deadlock-free multicast routing method is proposed for 3D mesh-based NoCs without using virtual channels. Unlike conventional turn models which prohibit certain turns to avoid deadlock, the proposed scheme restricts the locations where certain turns can take place to provide a more even degree of routing adaptiveness. Moreover, different sets of rules are employed for different layers of the NoC to decrease the likelihood of congestion in the network. The simulation results demonstrate the efficiency of the proposed method due to a balanced traffic distribution across the network.