We report the results of a molecular dynamics study of the effect of texture on the yield and peak stresses in columnar-grained nanotwinned Ag and Cu. The simulations suggest that in pure nanotwinned face-centered cubic metals, the strength is determined primarily by the cooperation or competition between two major factors: the magnitude of the Schmid factors for the available slip systems and the effectiveness of grain boundaries (and their triple-junctions) in generating dislocations. These factors and their relative impact depend on the geometry of the specimen relative to the applied stress, which is typically reflected in the texture of the material in experimental studies. The detailed mechanisms of plastic deformation are discussed for seven specific geometries that represent a range of different textures.