We investigate the dynamics of interacting magnetic flux lines driven by an external current in the presence of linear pinning centers, arranged either in a periodic square lattice or placed randomly in space, by means of three-dimensional Monte Carlo simulations. Compared to the non-interacting case, the repulsive forces between the vortices reduce the critical current J_c, as determined from the depinning threshold in the current-voltage (I-V) characteristics. Near the depinning current J_c, the voltage power spectrum S(w) reveals broad-band noise, characterized by a 1/wa power law decay with a <= 2. At larger currents the flux lines move with an average velocity v_{cm}. For a periodic arrangement of columnar pins with a lattice constant d and just above J_c, distinct peaks appear in the voltage noise spectrum S(w) at w ~ v_{cm}/d which we interpret as the signature of stick-slip flux line motion.