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A device capable of converting single quanta of the microwave field to the optical domain is an outstanding endeavor in the context of quantum interconnects between distant superconducting 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 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 strain-optical effects. Its efficiency results from the combination of integrated Si3N4 photonic circuits with ultra-low loss sustaining high intracavity photon numbers 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 has 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 near quantum coherent transduction is possible, with realistic experimental parameters. Combined with the high thermal conduction via the device bulk, heating effects are mitigated, and the approach does not require superconducting resonators that are susceptible to absorption of optical photons.