Nonlinear dynamics of an optically injected semiconductor laser is applied to photonic microwave generation. By properly adjusting the injection conditions, the optical frequency of the slave laser is first locked to the master laser. The slave laser is then driven into a periodic dynamical state, resulting in a single-sideband (SSB) microwave modulation on the optical carrier. The frequency of the SSB can be controlled by the optical injection strength and detuning. Frequency-modulated SSB can, thus, be obtained from a modulated injection. In this work, we experimentally investigate the generated SSB in terms of its broad tunability and fast modulation response. The results suggest application of this system in radio-over-fiber and optical subcarrier multiplexing technologies when microwave frequency modulation or frequency-shift keying is employed.