Methylammonium lead iodide perovskite can make high-efficiency solar cells, which also show an unexplained photocurrent hysteresis dependent on the device-poling history. Here we report quasielastic neutron scattering measurements showing that dipolar CH3NH3(+) ions reorientate between the faces, corners or edges of the pseudo-cubic lattice cages in CH3NH3PbI3 crystals with a room temperature residence time of ∼14 ps. Free rotation, π-flips and ionic diffusion are ruled out within a 1-200-ps time window. Monte Carlo simulations of interacting CH3NH3(+) dipoles realigning within a 3D lattice suggest that the scattering measurements may be explained by the stabilization of CH3NH3(+) in either antiferroelectric or ferroelectric domains. Collective realignment of CH3NH3(+) to screen a device's built-in potential could reduce photovoltaic performance. However, we estimate the timescale for a domain wall to traverse a typical device to be ∼0.1-1 ms, faster than most observed hysteresis.