We propose an asynchronous parallel algorithm for the linear Boltzmann transport equation in three dimensions, for Boron Neutron Capture Therapy (BNCT) radiation therapy planning, which can be implemented efficiently on shared memory parallel computers. The three-dimensional multigroup discrete ordinates transport equation is cast into a set of coupled two-dimensional equations, and discretization is accomplished on a grid of arbitrarily-shaped prismatic cells. This allows the 3D multigroup discrete ordinates transport equation to be solved using spatial parallel techniques originally developed in a recent work for the 2D discrete ordinates transport equation by using domain decomposition techniques on an unstructured triangular mesh. We use the Single Program Multiple Data approach to formulate a shared memory parallel numerical implementation with OpenMP. Our results demonstrate that the parallel version of the 3D deterministic algorithm yields good parallel efficiency.