A novel approach to generating 40 GHz narrow linewidth frequency-switched microwave signals is proposed and demonstrated. In this scheme, a single-longitudinal-mode (SLM) double-Brillouin-frequency spaced Brillouin fiber laser with dual-ring configuration and unpumped erbium-doped fiber (EDF) is used to generate dual-wavelength lasers, and a fiber Bragg grating is used to select the laser for different Brillouin frequency spacings. Dual-ring configuration and unpumped EDF are designed to select the mode for the SLM laser. Dual-wavelength lasers are inserted into a photodetector, and microwave signals at 10.66, 21.39, 32.12, or 42.85 GHz can be obtained. The linewidth of the generated microwave signals is less than 69 kHz. The frequency drift at each frequency is less than 0.83 MHz. The frequency noise and linewidth of Stokes signals are measured, and the linewidth broadening effect of microwave signals is analyzed.