We investigate driven magnetic flux lines in layered type-II superconductors subject to various configurations of strong point or columnar pinning centers by means of a three-dimensional elastic line model and Metropolis Monte Carlo simulations. We characterize the resulting nonequilibrium steady states by means of the force-velocity / current-voltage curve, static structure factor, mean vortex radius of gyration, number of double-kink and half-loop excitations, and velocity / voltage noise spectrum. We compare the results for the above observables for randomly distributed point and columnar defects, and demonstrate that the three-dimensional flux line structures and their fluctuations lead to a remarkable variety of complex phenomena in the steady-state transport properties of bulk superconductors.