A scheme for photonic generation of linearly chirped microwave waveforms (LCMWs) with a large time-bandwidth product (TBWP) is proposed and demonstrated based on an optically injected semiconductor laser. In the proposed system, the optically injected semiconductor laser is operated in period-one (P1) oscillation state. After optical-to-electrical conversion, a microwave signal can be generated with its frequency determined by the injection strength. By properly controlling the injection strength, an LCMW with a large TBWP can be generated. The proposed system has a simple and compact structure. Besides, the center frequency, bandwidth, as well as the temporal duration of the generated LCMWs can be easily adjusted. An experiment is carried out. LCMWs with TBWPs as large as 1.2x105 (bandwidth 12 GHz; temporal duration 10 μs) are successfully generated. The flexibility for tuning the center frequency, bandwidth and temporal duration is also demonstrated.