We describe progress toward a precise measurement of the recoil energy of an atom measured using a perturbative grating-echo atom interferometer (AI) that involves three standing-wave (sw) pulses. With this technique, a perturbing sw pulse is used to shift the phase of excited momentum states---producing a modulation in the contrast of the interference pattern. The signal exhibits narrow fringes that revive periodically at twice the two-photon recoil frequency, 2ω_q, as a function of the onset

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... ime of the pulse. Experiments are performed using samples of laser-cooled rubidium atoms with temperatures ≲ 5 μK in a non-magnetic apparatus. We demonstrate a measurement of ω_q with a statistical uncertainty of 37 parts per 10^9 (ppb) on a time scale of ∼ 45 ms in 14 hours. Further statistical improvements are anticipated by extending this time scale and narrowing the signal fringe width. However, the total systematic uncertainty is estimated to be ∼ 6 parts per 10^6 (ppm). We describe methods of reducing these systematic errors.