In this paper, we propose DANDi (Dynamic Asynchronous Neighbor Discovery Protocol for Directional Antennas), a neighbor discovery protocol for Wireless Sensor Networks (WSN) with directional antennas that guarantees that every communication link in a network is discovered. DANDi is asynchronous, fully directional (supports both directional transmissions and receptions) and has a dynamic contention resolution mechanism so no network topology information is needed in advance. It was implemented in Contiki, an open-source operating system for WSN and the Internet of Things, and extensively tested using the COOJA network simulator with Tmote Sky nodes equipped with 6-sectored antennas. The neighbor discovery times are deeply analyzed and analytical expressions for these times are presented. The DANDi protocol performance is assessed through simulations and compared with SAND, the state of the art protocol for this kind of networks. Our experiments based on simulations show that the discovery time is reduced 19% for networks with no collisions, and more than four times in average for unevenly dense networks. To the best of our knowledge, DANDi is faster than any other protocol in the state of the art with the great advantage of being able to discover every node in a network without requiring any prior information.