Active microelectromechanical systems (MEMS) with integrated electronic sensing and actuation can provide fast and sensitive measurements of force, acceleration and biological analytes. Strain sensors integrated onto MEMS cantilevers are widely used to transduce an applied force to an electrical signal in applications like atomic force microscopy and molecular detection. However, the high Young's moduli of traditional MEMS materials, such as silicon or silicon nitride, limit the thickness of the devices and, therefore, the deflection sensitivity that can be obtained for a specific spring constant. Here, we show that polymer materials with a low Young's modulus can be integrated into polymer-semiconductor-ceramic MEMS cantilevers that are thick and soft. We develop a multi-layer fabrication approach so that high-temperature processes can be used for the deposition and doping of piezoresistive semiconductor strain sensors without damaging the polymer layer. Our trilayer cantilever exhibits a sixfold reduction in force noise compared to a comparable self-sensing silicon cantilever. Furthermore, the strain-sensing electronics in our system are embedded between the polymer and ceramic layers, which makes the technology fluid-compatible.