We demonstrate a terahertz (THz) emitter based on Pt/BiFeO3 heterostructures, leveraging the ultrafast strain-wave-driven spin dynamics for coherent THz generation. Femtosecond laser pulses absorbed in the Pt layer generate elastic strain waves that propagate into the epitaxial BiFeO3 thin films, deflecting the antiferromagnetic Néel vector and exciting out-of-plane spin-wave (magnons) via magnetostriction. These spin waves are converted into ultrafast charge currents through the inverse spin Hall effect in Pt, and then the charge currents excite significant THz radiations. The excitation mechanism, confirmed by polarization-independent emission and a cosine azimuthal dependence, excludes optical nonlinearities and highlights the importance of the interface strain. While electric-field control of THz emission was not experimentally realized, this work establishes Pt/BiFeO3 as a promising platform for nonvolatile, electrically tunable spintronic THz emitters, exploiting the inherent magnetoelectric coupling of multiferroic BiFeO3.