A method is presented for determining the trajectory of a particle moving along a straight line orbit in axisymmetric toroidal coordinate systems. These geometries occur in problems associated with neutral, neutron, and radiation transport studies in tokamak fusion devices. Numerical solutions of the equations describing the trajectory are performed in geometric configurations generated by the solution of the plasma equilibrium problem. An example problem of the deposition of a pencil beam of high-energy neutral particles in a tokamak plasma with a noncircular cross section due to the necessity of incorporating a divertor and the desirability of operating at a high plasma energy density is described.