We have developed two configurations of an echo interferometer that rely on standing wave excitation of a laser-cooled sample of rubidium atoms. Both configurations can be used to measure acceleration a along the axis of excitation. For a two-pulse configuration, the signal from the interferometer is modulated at the recoil frequency and exhibits a sinusoidal frequency chirp as a function of pulse spacing. In comparison, for a three-pulse stimulated echo configuration, the signal is observed without recoil modulation and exhibits a modulation at a single frequency as a function of pulse spacing. The three-pulse configuration is less sensitive to effects of vibrations and magnetic field curvature leading to a longer experimental timescale. For both configurations of the atom interferometer (AI), we show that a measurement of acceleration with a statistical precision of 0.5% can be realized by analyzing the shape of the echo envelope that has a temporal duration of a few microseconds. Using the two-pulse AI, we obtain measurements of acceleration that are statistically precise to 6 parts per million (ppm) on a 25 ms timescale. In comparison, using the three-pulse AI, we obtain measurements of acceleration that are statistically precise to 0.4 ppm on a timescale of 50 ms. A further statistical enhancement is achieved by analyzing the data across the echo envelope so that the statistical error is reduced to 75 parts per billion (ppb). The inhomogeneous field of a magnetized vacuum chamber limited the experimental timescale and resulted in prominent systematic effects. Extended timescales and improved signal-to-noise ratio observed in recent echo experiments using a non-magnetic vacuum chamber suggest that echo techniques are suitable for a high precision measurement of gravitational acceleration g. We discuss methods for reducing systematic effects and improving the signal-to-noise ratio. Simulations of both AI configurations with a timescale of 300 ms suggest that an optimized experiment with improved vibration isolation and atoms selected in the mF = 0 state can result in measurements of g statistically precise to 0.3 pbb for the two-pulse AI and 0.6 ppb for the three-pulse AI.