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This paper studies flapping-wing micro air vehicles (FWMAV) whose transmission mechanisms use flexures as energy storage elements to reduce needed input power. A distinguishing feature of the proposed four-bar mechanism is the use of rubber-based flexures in two of its joints. These lightweight and compact flexures have been used for the first time in the design of an FWMAV whose projected total weight is approximately 3 g. This paper discusses in detail how the flexures were designed and how the challenges associated with their fabrication were met. Flexure stiffnesses were chosen based upon a simple, computationally efficient model of the four-bar mechanism actuated by an electric motor to flap two wings at 18 Hz. An instrumented test stand was designed to easily replace the upper part of the four-bar flexure mechanism and wings, and it was used to experimentally determine the power savings associated with flexures of different stiffnesses. While the measured power savings (maximum of 20%) may seem modest, they were nevertheless significant, considering that the use of the rubber-based flexures produced approximately 0.3 g added thrust at a less than 1% cost in weight (0.02 g).