We investigate a large class of linear boundary value problems for the general first-order one-dimensional hyperbolic systems in the strip $[0,1]\times\R$. We state rather broad natural conditions on the data under which the operators of the problems satisfy the Fredholm alternative in the spaces of continuous and time-periodic functions. A crucial ingredient of our analysis is a non-resonance condition, which is formulated in terms of the data responsible for the bijective part of the Fredholm operator. In the case of $2\times 2$ systems with reflection boundary conditions, we provide a criterium for the non-resonant behavior of the system.

18 pages