In the context of atomistic simulations of solids, two situations often occur: that in which disordered structures (e.g. from "amorphous cell" simulations) are deemed to occur with essentially equal likelihood to form a "glass", and that in which a particular orientational average over one crystal unit cell is desired, e.g. when a fiber modulus is deduced surmising that identical crystallites are oriented in the direction of the fiber axis with a specified direction of the unit cell frame while all orientations in the transverse directions are equally likely ("fiber symmetry"). The common averaging of elastic constants yields inappropriate results. We apply methods introduced by Hill and by Walpole more than three decades ago and show that with these methods, physically reasonable, self-consistent averages for elastic constants can be obtained as well as bounds considerably narrower than the well-known ones after Voigt and Reuss.