Calculation of neutron flux in three-dimensions is a complex problem. A novel approach for solving complicated neutron transport problems is presented, based on the tree multigrid technique and the Simplified P3 (SP3) approximation. Discretization of the second-order elliptic SP3 equations is performed for a regular three-dimensional octree grid by using an integrated scheme. The octree grid is generated directly from STL files, which can be exported from practically all computer-aided design-systems. Marshak-like boundary conditions are utilized. Iterative algorithms are constructed for SP3 approximation with simple coarse-to-fine prolongation and fine-to-coarse restriction operations of the tree multigrid technique. Numerical results are presented for a simple cylindrical homogeneous one-group test case and for a simplistic two-group pressurized water reactor pressure vessel fluence calculation benchmark. In the former homogeneous test case, a very good agreement with 1.6% maximal deviation compared with DORT results was obtained. In the latter test case, however, notable discrepancies were observed. These comparisons show that the tree multigrid technique is capable of solving three-dimensional neutron transport problems with a very low computational cost, but that the SP3 approximation itself is not satisfactory for all problems. However, the tree multigrid technique is a very promising new method for neutron transport.