We demonstrate the realization of a scheme for cold-atom gravimetry based on the recently demonstrated use of double-diffraction beam splitters [T. Leveque, A. Gauguet, F. Michaud, F. Pereira Dos Santos, and A. Landragin, Phys. Rev. Lett. 103, 080405 (2009)], where the use of two retro-reflected Raman beams allows symmetric diffraction in +-(Planck constant/2pi)k{sub eff} momenta. Although in principle restricted to the case of zero Doppler shift, for which the two pairs of Raman beams are simultaneously resonant, such diffraction pulses can remain efficient on atoms with nonzero velocity, such as in a gravimeter, when the frequency of one of the two Raman laser sources is modulated. Such pulses are used to realize an interferometer insensitive to laser phase noise and some of the dominant systematics. This approach reduces the technical requirements and would allow the realization of a simple atomic gravimeter. A sensitivity of 1.2x10{sup -7}g per shot is demonstrated.