In this paper, a multi-passband microwave photonic filter (MPF) based on multiple dispersive devices has been proposed and experimentally demonstrated. The Mach-Zehnder interferometer (MZI) divides the broadband light source (BBS) into multiple optical taps, and with the combination of different dispersion mediums such as chirped fiber Bragg grating (CFBG) and single mode fiber (SMF) to delay the optical tap, a MPF with multiple passbands can be simply achieved. The number of the passbands can be easily controlled by changing the number of the dispersion medium. In the experiment, the frequency response result of the four passbands is obtained by accessing two CFBGs and two SMFs. In addition, by adjusting the wavelength interval of the interference spectrum with a variable optical delay line (VODL), all passbands of the filter can be simultaneously tuned. The filter has broad application prospects in the fields of modern wireless and satellite communication, optoelectronic oscillator and optical sensing.