We demonstrate experimentally and theoretically the reverse Doppler effect when magnons with negative group velocity are reflected off a moving Bragg grating. This grating, which represents a moving magnonic crystal, is created in an yttrium-iron-garnet film by the periodic strain induced by a traveling surface acoustic wave. As reflection occurs from a crystal rather than from a single reflecting surface, the wave number of the scattered wave is strictly determined by a momentum conservation law. Magnons scattered from the approaching (receding) magnonic crystal are found to be shifted down (up) in frequency. This result, together with an earlier report of reverse Doppler shift from moving sources [ D. D. Stancil et al. Phys. Rev. B 74 060404 (2006)], establishes that the reverse Doppler effect is a universal phenomenon in systems with negative group velocity and not restricted to left-handed materials.