We characterize the temporal response of fiber-optic components using a fiber-based frequency comb interferometer; measurements are compared and validated against a commercial instrument. The main advantage of the instrument lies in the absence of moving parts or a tunable laser, leading to very fast scanning. A measurement of a mechanical distortion, cycled at 130 Hz, on a fiber Bragg grating (FBG) is presented. A complete profile of the mechanical distortion is taken every 2.5 ms (400 Hz scanning speed) and each "snapshot" is taken in 200 micros. This scanning speed was arbitrarily chosen, and the instrument could be set to scan much faster, up to hundreds of kilohertz. With high-reflectivity FBGs, the same instrument could scan simultaneously the profile of 140 wavelength-multiplexed FBGs at 2 kHz.