Neutral hydrogen transport in a fully ionized two-dimensional tokamak plasma was examined using discrete ordinates and contrasted with earlier analyses. In particular, curvature effects induced by toroidal geometries and ray effects caused by possible source localization were investigated. From an overview of the multigroup discrete-ordinates approximation, methodology in two-dimensional cylindrical geometry is detailed, mesh and plasma zoning procedures are sketched, and the piecewise polynomial solution algorithm on a triangular domain is obtained. Toroidal effects and comparisons as related to reaction rates and particle spectra are examined for various model and source configurations. For symmetric source distributions, toroidal effects on fluxes scale roughly as R/rj, with R the major axis and rj measured along the major toroidal axis. Increases in collision rates and decreases in leakages are also noted for the system. Effects on a sputtering model and measurement techniques for the charge exchange spectrum show that decrease in sputtered outflux due to geometry is exhibited with greatest asymmetry in sputtered flux along the major toroidal axis. Directional dependence of a plasma measurement technique is specifically linked to toroidal flux variations, with the result that lesser inner and greater outer wall temperatures are predicted. Ray effects for localized sources in the plasma are categorized and negated with fictitious source methods. It is found that isolated neutral sources cause ray effects only in highly homogeneous plasmas where ionization dominates charge exchange processes.