Fiber Bragg grating (FBG) accelerometers are extensively utilized across various industries. For a high-performance FBG accelerometer interrogator, achieving low cost, wide range, multi-channel capability, high precision, and high-speed demodulation is critical. This paper proposes a chip-level wavelength demodulation method for FBG accelerometers utilizing a cascaded micro-ring resonator (MRR) array. Capable of demodulating multiple accelerometers within a 27.8 nm bandwidth at a speed of up to 1.6 kHz, the system's performance is evaluated in terms of minimum detectable wavelength shift and noise level. We compare our system with a commercial FBG interrogator, and the results underscore the effectiveness and feasibility of our FBG accelerometer demodulation method, offering potential implications for various applications.