Geographic routing is an attractive localized routing scheme for wireless sensor networks (WSNs) due to its desirable scalability and efficiency. Maintaining neighborhood information for packet forwarding can achieve a high efficiency in geographic routing, but may not be appropriate for WSNs in highly dynamic scenarios where network topology changes frequently due to nodes mobility and availability. We propose a novel online routing scheme, called Energy-Efficient Beaconless Geographic Routing (EBGR), which can provide loop-free, fully stateless, energy-efficient sensor-to-sink routing at a low communication overhead without the help of prior neighborhood knowledge. In EBGR, each node first calculates its ideal next-hop relay position on the straight line towards the sink based on the energy optimal forwarding distance, and each forwarder selects the neighbor closest to its ideal next-hop relay position as the next-hop relay using the RTS/CTS (Request-To-Send/Clear-To-Send) handshaking mechanism. We establish the lower and upper bounds on hop count and the upper bound on energy consumption under EBGR for sensor-to-sink routing, assuming no packet loss and no failures in greedy forwarding. Moreover, we demonstrate that the expected total energy consumption along a route towards the sink under EBG approaches to the lower bound with the increase of node deployment density. We also extend EBGR to lossy sensor networks to provide energy-efficient routing in the presence of unreliable communication links. Simulation results show that our scheme significantly outperforms existing protocols in wireless sensor networks with highly dynamic network topologies.