The smoothed particle hydrodynamics (SPH) technique has been applied to a problem in kinetic theory, namely, the dynamics of liquid crystalline polymers (LCPs). It is a Lagrangian solution method developed for fluid flow calculations; its adaption to kinetic theory is outlined. The Lagrangian formulation of the Doi theory for LCPs is first described, and the problem is presented in the general framework of nonparametric density estimation. The implementation of the SPH technique in this specific problem is given, highlighting particular aspects of our implementation of SPH, including the form of the kernel function and use of an adaptive kernel. We then present results which demonstrate convergence and other details of the solution method, and also make comparisons with other solution techniques and discuss other potential applications.