qubits, but likewise can have applications in other fields, such as radio astronomy or, in the classical realm, microwave photonics. A variety of transduction approaches, based on optomechanical or electro-optical interactions, have been proposed and realized, yet the required vanishing added noises and an efficiency approaching unity, have not yet been attained. Here we present a new transduction scheme that could in theory satisfy the requirements for quantum coherent bidirectional transduction. Our scheme relies on an intermediary mechanical mode, a high overtone bulk acoustic resonance (HBAR), to coherently couple microwave and optical photons through the piezoelectric and strainoptical effects. Its efficiency results from ultra low loss and high intracavity photon number sustaining integrated silicon nitride photonic circuits, combined with the highly efficient microwave to mechanical transduction offered by piezoelectrically coupled HBAR. We develop a quantum theory for this multipartite system by first introducing a quantization method for the piezoelectric interaction between the microwave mode and the mechanical mode from first principles (which to our knowledge had not been presented in this form), and link the latter to the conventional Butterworth-Van Dyke model. The HBAR is subsequently coupled to a pair of hybridized optical modes from coupled optical ring cavities via the strain-optical effect. We analyze the conversion capabilities of the proposed device using signal flow graphs, and demonstrate that quantum coherent transduction is possible, with realistic experimental parameters at optical input laser powers of the order of O(mW).