In this paper, the disturbance attenuation problem is formulated and solved for a class of linear hybrid systems in the presence of periodic jumps. The results are achieved, both in the finite and infinite horizon cases, by borrowing ideas from the theory of dynamic games. In the considered formulation, independent disturbances affecting the continuous-time and the discrete-time components of the hybrid system are allowed. Moreover, the analysis is carried out by introducing easily verifiable conditions, involving the definition of a Monodromy Riccati Equation, i.e., a classical Riccati equation defined for the one-period discrete-time equivalent model. Interestingly, as a by-product, the main statements essentially characterize the solution of zero-sum noncooperative dynamic games for periodic linear hybrid systems, which is of interest per se.