Anypath routing is proposed to improve the performance of wireless networks by exploiting the spatial diversity and broadcast nature of the wireless medium. In this paper, we study anypath routing in wireless networks with directional antennas, and propose DART (Directional Anypath RouTing), a crosslayer design of MAC and routing layers. For the routing layer, we propose two shortest directional anypath routing algorithms based on two antenna models. Our routing algorithms are simple and fast, thus are suitable for implementation in practical protocols. For the MAC layer, we present a directional anycast MAC, which is an enhancement to the IEEE 802.11 MAC, making DART suitable for integration into current systems. Simulation results show that DART can significantly reduce the packet transmission delay.