A type of “phase-space discontinuous diamond” difference scheme, or “phase-space linear discontinuous finite element” approximation, is implemented to solve the two-dimensional [(x-y) or (r-z)] neutron transport equation. The results obtained on some well-known transport benchmark problems are much more accurate than discrete ordinates solutions attained with spatial diamond differencing or discontinuous finite element approximations. Error studies show convergence to the phase-space fine-mesh limit solution with an approximate and convergence rate, at least in the case of rectangular cells on phase-space domain D × V. In addition, phase-space fine-mesh limit results have been estimated with the help of extrapolation procedures for some neutron transport benchmark problems. This phase-space linear discontinuous finite element approach can be easily enlarged to more general spaces.