Based on the kinetic theory of photons, a new lattice Boltzmann method for the simulation of 3D radiation transport is presented. The method was successfully validated with Monte Carlo simulations of radiation transport in optical thick absorbing and non-absorbing turbid media containing either isotropic or anisotropic scatterers. Moreover, for the approximation of Mie-scattering, a new iterative algebraic approach for the discretization of the scattering phase function was developed, ensuring full conservation of energy and asymmetry after discretization. It was found that the main error sources of the method are caused by linearization and ray effects and suggestions for further improvement of the method are made.