We demonstrate a fiber Bragg grating (FBG)-based oscillator-accelerometer in which the acceleration sensitivity can be tuned by controlling the location of the mass oscillator. We theoretically and experimentally investigated the performance of the proposed accelerometer. Theoretical analysis showed that both the mass and location of the oscillator affect the sensitivity and resonant frequency of the accelerometer. To simplify the analysis, a nondimensional parameter, P, was introduced to tune the sensitivity of the FBG-based oscillator-accelerometer, which is related to the location of the mass oscillator. Numerical analysis showed that the accelerometer sensitivity is linearly proportional to the P parameter. In the experiment, six FBG-based oscillator-accelerometers with different P parameters (0.125, 0.25, 0.375, 0.5, 0.625, 0.75) were fabricated and tested. The experimental results agree very well with the numerical analysis, in which the sensitivity of the proposed accelerometer linearly increased with the increase in parameter P (7.6 pm/g, 15.8 pm/g, 19.3 pm/g, 25.4 pm/g, 30.6 pm/g, 35.7 pm/g). The resonance frequency is quadratically proportional to parameter P, and the resonance frequency reaches the minimum of 440 Hz when P is equal to 0.5. The proposed oscillator-accelerometer showed very good orthogonal vibration isolation.