In classical routing protocols, geographical distances/locations are typically used as the metric to select the best route, under the assumption that shorter distances exhibit lower energy consumption and nodes within the communication range of the sender can receive packets with a certain success probability. However, in underwater acoustic sensor networks (UASNs), sound propagation in the ocean medium is more complex than that in the air due to many factors, including sound speed variations and the interaction of sound waves with the sea surface and floor, causing the sound rays to bend. Therefore, propagation of sound is anisotropic in water, and may cause a phenomenon called shadow zone where nodes in the communication range of the sender cannot hear any signal. This renders conventional routing protocols no longer energy-efficient. In this paper, we make use of the ray-model to account for the environment-dependent behavior of the underwater channel, re-define nodes' one-hop neighbors based on signal attenuation rather than geographical distance, and design a distributed energy-efficient routing protocol for UASNs. Results show that our ray-model-based routing policy consistently outperforms the shortest path policy, and performs very close to the optimal one in several scenarios.