Within the framework of functional data analysis, we develop principal component analysis for periodically correlated time series of functions. We define the components of the above analysis including periodic, operator-valued filters, score processes and the inversion formulas. We show that these objects are defined via convergent series under a simple condition requiring summability of the Hilbert-Schmidt norms of the filter coefficients, and that they poses optimality properties. We explain how the Hilbert space theory reduces to an approximate finite-dimensional setting which is implemented in a custom build R package. A data example and a simulation study show that the new methodology is superior to existing tools if the functional time series exhibit periodic characteristics.