The simplified P3 (SP3) approximation to the multigroup neutron transport equation in arbitrary geometries is derived using a variational analysis. This derivation yields the SP3 equations along with material interface and Marshak-like boundary conditions. The multigroup SP3 approximation is reformulated as a system of within-group problems that can be solved iteratively. An "explicit" iterative algorithm for solving the within-group problem is described, Fourier analyzed, and shown to be more efficient than the traditional FLIP implicit algorithm. Numerical results compare diffusion (P1), simplified P2 (SP2), and simplified P3 calculations of a mixed-oxide (MOX) fuel benchmark problem to a reference transport calculation. The SP3 approximation can eliminate much of the inaccuracy in the diffusion and SP2 calculations of MOX fuel problems.