Surface wave studies of the Reykjanes Ridge (RR) and the Iceland hotspot have imaged an unusual and enigmatic pattern of two zones of negative radial anisotropy on each side of the RR. We test previously posed and new hypotheses for the origin of this anisotropy, by considering lattice preferred orientation (LPO) of olivine A-type fabric in simple models with 1-D, layered structures, as well as in 2-D and 3-D geodynamic models with mantle flow and LPO evolution. Synthetic phase velocities of Love and Rayleigh waves traveling parallel to the ridge axis are produced and then inverted to mimic the previous seismic studies. Results of 1-D models show that strong negative radial anisotropy can be produced when olivine a axes are preferentially aligned not only vertically but also subhorizontally in the plane of wave propagation. Geodynamic models show that negative anisotropy on the sides of the RR can occur when plate spreading impels a corner flow, and in turn a subvertical alignment of olivine a axes, on the sides of the ridge axis. Mantle dehydration must be invoked to form a viscous upper layer that minimizes the disturbance of the corner flow by the Iceland mantle plume. While the results are promising, important discrepancies still exist between the observed seismic structure and the predictions of this model, as well as models of a variety of types of mantle flow associated with plume-ridge interaction. Thus, other factors that influence seismic anisotropy, but not considered in this study, such as power-law rheology, water, melt, or time-dependent mantle flow, are probably important beneath the Reykjanes Ridge.