Signal detection and processing have become an important way to diagnose mechanical faults. The classical bistable stochastic resonance (CBSR) method for signal detection can become saturated, where the amplitude of the output signal gradually stabilizes at a relatively low level instead of increasing with further increases of the input signal amplitude. This leads to difficulty in extracting the weak signals from strong background noise. We studied a new mechanism based on unsaturated piecewise linear stochastic resonance (PLSR). The piecewise linear potential model has a unique structure, which can independently adjust the barrier height and potential wall inclination, so the piecewise linear potential model has a rich potential structure. The rich potential structure makes the potential model unsaturated, thus ensuring that the output signals increase as the input signals increase. In addition, according to the piecewise linear model, the output signal-to-noise ratio (SNR) of the system is deducted. Analysis of the influence of signal strength, potential parameters, and angular frequency on the SNR shows that the optimal SNR can be obtained by adjusting the potential parameters. We propose a weak signal detection method for bearing fault diagnosis. This method can effectively extract the weak fault signals from rolling bearings in a strong noise background. The simulated and experimental bearing fault signals verify that the proposed PLSR method is superior to the CBSR method.