Using Lagrangian traces of particle trajectories, the deposition of ellipsoidal particles in a fully developed turbulent channel flow is analysed. The turbulent velocity field is simulated by a continuous Gaussian random vector field superimposed on an empirical mean velocity profile. Particle trajectories are evaluated using the equations of motion for a rigid ellipsoid translating and rotating in a turbulent flow field. An implicit integration scheme is used for calculating particle velocities induced by drag and gravity. Euler's four parameters are used for describing the particle orientation. The governing equations for the translational and rotational motion of particles are outlined, and simulation results for ellipsoidal particles with minimum diameters from 1 to 50/~m and aspect ratios from one to ten are presented. The cases of vertical and horizontal ducts are studied. Good agreement of the predicted deposition velocities with the experimental data and the empirical equation is observed. For a fixed minimum diameter and/or fixed equivalent relaxation time, the results show that the deposition velocity increases rapidly with aspect ratio. When stated in terms of the relaxation time based on the equivalent volume sphere, the deposition rate of ellipsoidal particles is found not to vary appreciably with changes in aspect ratio.