A fiber Bragg grating (FBG) sensor using wide and fast mode-locked wavelength-swept laser is demonstrated. The key device is the optical source, wide and fast wavelength-tunable mode-locked fiber laser based on dispersion tuning. This laser has wide tuning range (over 100nm) and a fast tunable speed (up to 20 kHz). Hence, it is suited for the dynamic FBG sensor system. Our sensor system enables dynamic measurement of strain in real-time at a high measurement rate. In this system,the signals of the sensor are obtained by analyzing temporal waveforms of refleceted light from the FBG without observing the spectrum. A measurement speed as high as 1 kHz is achieved when the wavelength-tunable source is swept at 1 kHz rate. The stability (indicated by the standard deviations of the results) is 30-60 pm in optical spectrum.