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Vehicular ad hoc networks (VANETs) are the key enabling technology for intelligent transportation systems. Carrier-sense multiple access with collision avoidance (CSMA/CA) is the de facto media access standard for inter-vehicular communications, but its performance degrades in high-density networks. Time-division multiple access (TDMA)-based protocols fill this gap to a certain extent, but encounter inefficient clock synchronization and lack of prioritized message delivery. Therefore, we propose a priority-based direction-aware media access control (PDMAC) as a novel protocol for intra-cluster and inter-cluster clock synchronization. Furthermore, PDMAC pioneers a three-tier priority assignment technique to enhance warning messages delivery by taking into account the direction component, message type, and severity level on each tier. Analytical and simulation results validate the improved performance of PDMAC in terms of clock synchronization, channel utilization, message loss rate, end-to-end delays, and network throughput, as compared with eminent VANET MAC protocols.