Seismic body-wave and normal-mode data(1-4) suggest that the Earth’s solid inner core is elastically anisotropic, with the fast direction nearly parallel to the rotation axis, Compressional body-wave data also suggest that the anisotropy increases with turning depth, with a maximum anisotropy of 3-4% (refs 5-7). Here I propose that the inner core’s elastic anisotropy and the depth dependence of the anisotropy may be due to solidification texturing that results from the dendritic growth of iron crystals. I demonstrate through laboratory measurements that directionally solidified metallic alloys can indeed exhibit a significant elastic anisotropy due to solidification texturing. The anisotropy is due to the dendrites growing along a particular crystallographic axis(8,9), which tends to be aligned along the direction of heat flow. Directional cooling in the Earth’s core must therefore be predominantly in the cylindrically radial direction in order for solidification texturing to cause the observed anisotropy; such directional heat flow may be consistent with the pattern of convection in the outer core(10). It is possible that columnar crystals composed of dendrites and elongated in the cylindrically radial direction could also explain observations of inner-core attenuation anisotropy(11,12).