Radiation transport equations arise in the study of many different fields, such as combustion, astrophysics and hypersonic flow. The solution of these equations presents interesting challenges due to large jumps in the coefficients and strong nonlinearities. The simulation of radiation transport in the optically thick flux-limited diffusion regime has been identified as one of the most time-consuming tasks within many large simulation codes. Due to multimaterial complex geometry, the radiation transport system must often be solved on unstructured grids. Nonconforming finite element methods have proven flexible and effective on incompressible fluid flow problems. In this paper, we investigate the behavior and the benefits of the unstructured P_1 nonconforming finite element method in solving unsteady implicit nonlinear radiation diffusion problems in two dimensions, using various linearization methods. We consider a Newton method, a Picard method, and an Inexact Newton method. To solve the large-scale linear problems that arise, we use multigrid. Due to the nonnestedness of nonconforming finite element spaces, some difficulties arise, such as failure in solving coarsest level problem and the requirement of many smoothings. To overcome these difficulties, we use new intergrid transfer operators and the GMRES method with a V-cycle multigrid preconditioner.