A wideband true-time-delay (TTD) unit employing a novel fiber grating prism (FGP) for phased array beamforming is proposed in this paper. The FGP consists of a single-mode fiber delay line, a chirped grating delay line and three discrete fiber Bragg grating (FBG) delay lines. The first delay line is a length of single-mode fiber, which provides a fixed time delay for all wavelengths. The second delay line, which employs a chirped grating, is used to provide small time delays. The discrete FBG delay lines, which incorporate an array of 13 discrete FBGs, are used to provide large time delays. A 5x13 fiber-grating prism is constructed and experimented. The results show that the time delays produced by the fiber grating delay lines are independent of the microwave frequency and agree well with the calculated time delays. Based on the measured time delays, the radiation patterns of 5-element array antenna are calculated and analyzed. The beampointing direction of the antenna is independent of the microwave frequency and can be controlled by tuning the wavelength of the optical carrier. This 5x13 TTD unit can provide phased array beamforming at microwave frequencies up to 6 GHz.