A method is developed for obtaining solutions to the Boltzmann neutron transport equation on irregular triangular grids with nonorthogonal boundaries and anisotropic scattering. A functional is developed from the canonical form of the multigroup transport equation. The angular variable is then removed by expanding the functional in spherical harmonics, retaining only the first two flux moments and limiting the scattering to be linearly anisotropic. The finite element method is then implemented using quadratic Lagrange-type interpolating polynomials to span the spatial domain. The resultant set of coupled linear equations is then solved iteratively using the block successive over-relaxation method. A number of numerical experiments are performed to evaluate the performance of the proposed method. The results are compared to the results obtained by various established methods. In all cases, aggrement is excellent.