In this paper, we explore the effect of particle shape angularity on the mechanical behavior of sheared granular packings. A first series of contact dynamics simulations is performed in 2D with regular polygons with an increasing number of sides ranging from 3 (triangles) to 60. Then, in order to approach “idealized” angular particles, a second series of simulations is performed in 3D with irregular polyhedra with the number of faces ranging from 8 (octahedron-like) to 596. A counterintuitive finding is that the shear strength increases with angularity up to a maximum value and saturates as the particles become more angular (below 6 sides in 2D and 46 faces in 3D). A micromechanical analysis of force and contact orientations, all enhanced by face-face and face-side contacts, reveals that this increase is due to an increase of both contact and force anisotropies, and the saturation for higher angularities is a consequence of a rapid fall-off of the contact and normal force anisotropies compensated by an increase of the tangential force anisotropy.