Collective excitations in a fluxon chain placed in a periodically modulated Josephson junction are studied analytically and numerically. In order to eliminate fluxon collisions with boundaries, we consider a Josephson ring (annular Josephson junction). Due to the interaction of the fluxons with periodically placed obstacles, we predict that linear deformation modes of the fluxon chain should bring about resonances which be be observed experimentally. The linear analysis is compared with numerical simulations, and good agreement is found in an appropriate parameter range. In the ‘‘relativistic’’ limit, the numerical simulations reveal a dynamical mode which is characterized by a strongly nonlinear interaction between the moving fluxons in the chain. A qualitative explanation of this regime is suggested by an extrapolation of the linear behavior.