We investigate fluxon dynamics in underdamped one-dimensional parallel arrays of small Josephson tunnel junctions. The current-voltage characteristics of the arrays show various resonant steps depending upon the temperature of the sample and the externally applied magnetic field. Experimental results on fluxon propagation in arrays are compared with that for continuous Josephson transmission lines fabricated on the same chip. By modeling fluxon dynamic states in a one-dimensional array as a propagation of kinks in a discrete sine-Gordon lattice, we find consistency between numerical results and the experimental data. This consistency indicates that the concept of fluxons as moving relativistic particles can be still used even for strongly discrete lines. However, two important classes of phenomena are found which do not have any counterparts in the continuum case. These are concerned with the data that we obtain in the short-wavelength limit (determined by line discreteness) and with damping requirements that are necessary in order to stabilize kink propagation.