An Acousto-Optic Gyroscope (AOG) consisting of a photonic integrated device embedded into two inherently matched piezoelectric surface acoustic wave (SAW) resonators sharing the same acoustic cavity is presented. This constitutes the first demonstration of a micromachined strain-based optomechanical gyroscope that uses the effective index of the optical waveguide due to the acousto-optic effect rather than conventional displacement sensing. The theoretical analysis comparing various photonic phase sensing techniques is presented and verified experimentally for the cases based on a Mach-Zehnder interferometer, as well as a racetrack resonator. This first prototype integrates acoustic and photonic components on the same lithium niobate on insulator (LNOI) substrate and constitutes the first proof of concept demonstration of the AOG. This approach enables the development of a new class of micromachined gyroscopes that combines the advantages of both conventional microscale vibrating gyroscopes and optical gyroscopes.