This work demonstrates the capabilities of an entangled photon-pair source at telecom wavelengths, based on a photonic integrated Si3N4 microresonator with monolithically integrated piezoelectric frequency tuning. Previously, frequency tuning of photon pairs generated by microresonators has only been demonstrated using thermal control, however these have limited actuation bandwidth and are not compatible with cryogenic environments. Here, the frequency-tunable photon-pair generation capabilities of a Si3N4 microresonator with a monolithically integrated aluminium nitride layer are shown. Fast-frequency locking of the microresonator to an external laser is demonstrated, with a resulting locking bandwidth orders of magnitude larger than reported previously using thermal locking. These abilities will have direct application in future schemes which interface such sources with quantum memories based on, e.g., trapped-ion or rare-earth-ion schemes.