A phase field model accounting for inhomogeneous and anisotropic elasticity has been developed to study the growth of acicular precipitates. The diffusion-controlled growth of an isolated precipitate is investigated in the presence of elastic stresses generated by anisotropic transformation eigenstrains in isothermal conditions. It is shown that elastic anisotropy qualitatively changes the growth mechanisms and that a stationary regime may be reached. In light of these conclusions, the elastic anisotropies of several metallic alloys featuring Widmanstätten structures are analyzed. The growth regimes predicted by our analysis are consistent with experimental observations; this strongly suggests the primary importance of elastic anisotropy in the formation of these microstructures.