Membrane sensors have been widely used in various fields owing to their multifunctionality and cost-effectiveness. However, few studies have investigated frequency-tunable membrane sensors, which could enable versatility in the face of different device requirements while retaining high sensitivity, fast response times, and high accuracy. In this study, we propose a device comprising an asymmetric L-shaped membrane with tunable operating frequencies for microfabrication and mass sensing applications. The resonant frequency could be controlled by adjusting the membrane geometry. To fully understand the vibration characteristics of the asymmetric L-shaped membrane, the free vibrations of the membrane are first solved by a semi-analytical treatment combining domain decomposition and variable separation methods. The finite-element solutions confirmed the validity of the derived semi-analytical solutions. Parametric analysis results revealed that the fundamental natural frequency decreases monotonically with the increase in length or width of the membrane segment. Numerical examples revealed that the proposed model can be employed to identify suitable materials for membrane sensors with specific frequency requirements under a given set of L-shaped membrane geometries. The model can also achieve frequency matching by changing the length or width of membrane segments given a specified membrane material. Finally, performance sensitivity analyses for mass sensing were carried out, and the results showed that the performance sensitivity was up to 0.7 kHz/pg for polymer materials under certain conditions.