A first principles theoretical approach shows that, at the density of the inner core, both hexagonal [hexagonal close-packed (hcp)] and cubic [face-centered-cubic (fcc)] phases of iron are substantially elastically anisotropic. A forward model of the inner core based on the predicted elastic constants and the assumption that the inner core consists of a nearly perfectly aligned aggregate of hcp crystals shows good agreement with seismic travel time anomalies that have been attributed to inner core anisotropy. A cylindrically averaged aggregate of fee crystals disagrees with the seismic observations.