Photonic transmission of microwave signals from a central office to remote base stations is a key functionality in broadband radio-over-fiber access networks. Because of chromatic dispersion, a strong fluctuation of the microwave power along fiber transmission happens to microwave-modulated optical carriers with double-sideband features. Therefore, optical single-sideband modulation characteristics are preferred. Direct modulation of a semiconductor laser is the simplest scheme for photonic microwave generation and transmission. However, the symmetric property of the laser in the modulation sideband intensity makes the scheme unattractive for radio-over-fiber applications. In this study, we apply the injection locking technique to the laser for optical single-sideband generation. Proper optical injection can drive the laser to the stable-locking dynamical state before entering the Hopf bifurcation. The field-carrier coupling of the injected laser is radically modified due to the dynamical interaction between the injection-shifted cavity resonance and the injection-imposed oscillation. Therefore, the relaxation resonance sidebands of the injected laser are considerably shifted in frequency and asymmetrically modified in intensity, the extent of which depends strongly on the injection condition. Under the range of our study, direct modulation of the injected laser can thus generate microwave signals that are broadly tunable up to 4 times its free-funning relaxation resonance frequency and are highly asymmetric up to 20 dB in modulation sidebands. The microwave frequency can be tuned over a broad range while keeping a similar level of modulation sideband asymmetry, or different levels of modulation sideband asymmetry can be obtained while keeping a similar microwave frequency. This adds the flexibility and re-configurability to the proposed system. No optical phase-locking electronics, no high driving voltages, and no narrow-bandwidth optical filters are necessary as in many other systems.