A method for making a micro-electro-mechanical systems (MEMS) vibrating structure is disclosed. The MEMS is supported by a MEMS anchor system and includes a single-crystal piezoelectric thin-film layer that has a specific non-standard crystal orientation, which may be selected to increase an electromechanical coupling coefficient, decrease a temperature coefficient of frequency, or both. The MEMS vibrating structure may have dominant lateral vibrations or dominant thickness vibrations. The single-crystal piezoelectric thin-film layer may include Lithium Tantalate or Lithium Niobate, and may provide MEMS vibrating structures with precise sizes and shapes, which may provide high accuracy and enable fabrication of multiple resonators having different resonant frequencies on a single substrate.