The design for a photonic microwave filter tunable in both bandwidth and operating frequency is proposed and experimentally demonstrated. The circuit is based on a single sideband modulator used in conjunction with two or more transmission fiber Bragg gratings (FBGs) cascaded in series. It is demonstrated that the optical filtering characteristics of the FBGs are instrumental in defining the shape of the microwave filter, and the numerical modeling was used to optimize these characteristics. A multiphase-shift transmission FBG design is used to increase the dynamic range of the filter, control the filter ripple, and maximize the slope of the filter skirts. Initial measurements confirmed the design theory and demonstrated a working microwave filter with a bandwidth tunable from approximately 2 to 3.5 GHz and an 18 GHz operating frequency tuning range. Further work is required to refine the FBG manufacturing process and reduce the impact of fabrication errors.