A method for simulating coupled electromagnetic and mechanical vibrations on arbitrarily shaped piezoelectric structures is presented. This method is based on weak forms and can be implemented in any finite-element-method software, allowing editable access to their definitions. No quasi-static approximation is imposed, meaning that magnetic fields generated by displacement currents within piezoelectric materials are captured, enabling the flow of electromagnetic energy inside and around structures containing such material to be accurately simulated. The method is particularly relevant to the design of piezoelectric antennas, resonators, and waveguides exploiting either bulk or surface-acoustic waves. The accuracy and capabilities of the method are demonstrated by simulating, in COMSOL Multiphysics, (i) a Rayleigh mode on the surface of Z-cut lithium niobate crystal and (ii) a torsional mode of a cylinder of lead zirconium titanate (PZT-5H) ceramic functioning as a micro-antenna.