Utilizing available channels to improve the network performance is one of the most important targets for the cognitive MAC protocol design. Using antenna technologies is an efficient way to reach this target. Therefore, in this paper, we propose a novel cognitive MAC protocol, called Polarization-based Long-range Communication Directional MAC Protocol (PLRC-DMAC), for Cognitive Ad Hoc Networks (CAHNs). The proposed protocol uses directional antennas to acquire better spatial reuse and establish long-range communication links, which can support more nodes to access the same channel simultaneously. Moreover, the PLRC-DMAC also uses polarization diversity to allow nodes in the CAHN to share the same channel with Primary Users (PUs). Furthermore, we also propose a Long-range Orientation (LRO) algorithm to orient the long-range nodes. Simulation results show that the LRO algorithm can accurately orient the long-range nodes, and the PLRC-DMAC can significantly increase the network throughput as well as reduce the end-to-end delay.