Discrete ordinates calculations are presumed to translate particles from cell to cell in the directions specified in the angular set. This should result in uncollided particles from a small source propagating through the spatial mesh in narrow beams in these directions. Accurate high-order angular quadratures presume accurately attenuated propagation in the intended directions. This work examines the ability of various spatial quadratures to propagate rays correctly. Some widely used methods are shown to fail at this fundamental task. Diamond-difference approximations introduce undamped lateral oscillations, resulting in severely unphysical flux representations. Nonlinear fixups can prevent negativity but do not correct the underlying failure to properly propagate rays. First-moment conserving schemes tend to be successful but can be degraded in performance by simplifying approximations that are often used. Characteristic schemes are shown to have significant advantages. New characteristic methods are developed here that are exact (in a certain sense) in propagating rays and that uncouple the calculation of adjacent spatial cells in the mesh sweep. This enables DO loops to be converted to DO INDEPENDENT loops, with obvious implications for vector and/or parallel implementations.