The rapid development of quantum information processors has accelerated the demand for technologies that enable quantum networking. One promising approach uses mechanical resonators as an intermediary between microwave and optical fields. Signals from a superconducting, topological, or spin qubit processor can then be converted coherently to optical states at telecom wavelengths. However, current devices built from homogeneous structures suffer from added noise and a small conversion efficiency. Combining advantageous properties of different materials into a heterogeneous design should allow for superior quantum transduction devices - so far these hybrid approaches have however been hampered by complex fabrication procedures. Here we present a novel integration method, based on previous pick-and-place ideas, that can combine independently fabricated device components of different materials into a single device. The method allows for a precision alignment by continuous optical monitoring during the process. Using our method, we assemble a hybrid silicon-lithium niobate device with state-of-the-art wavelength conversion characteristics.