The whirligig beetle, claimed to be one of the most energy-efficient swimmers in the animal kingdom, has evolved a series of propulsion strategies that may serve as a source of inspiration for the design of propulsion mechanisms for energy-efficient surface swimming. In this paper, we introduce a robot platform that was developed to test an energy-efficient propulsion mechanism inspired by the whirligig beetle. A propulsor-body-fluid interaction dynamics model is proposed, and based on this model, the propulsor flexural rigidity and beating patterns are optimized in order to achieve energy-efficient linear swimming and turning. The optimization results indicate that a propulsor with decreasing flexural rigidity enhances vortex shedding and improves thrust generation. It has also been found that an alternating asymmetrical beating sequence and optimal beating frequency of 0.71 Hz improves propulsion efficiency for linear swimming of the robot. The alternating beating of the outboard propulsors and the unfolded inboard propulsors working as brakes results in efficient turning with a smaller turning radius. Both simulation and experimental studies were conducted, and the results illustrate that decreasing flexural rigidity along the propulsor length, an oscillating body motion, and an S-shaped trajectory are critical for energy-efficient propulsion of the robot.