Approximation properties of periodic quasi-projection operators with matrix dilations are studied. Such operators are generated by a sequence of functions φ_j and a sequence of distributions/functions φ˜_j. Error estimates for sampling-type quasi-projection operators are obtained under the periodic Strang-Fix conditions for φ_j and the compatibility conditions for φ_j and φ˜_j. These estimates are given in terms of the Fourier coefficients of approximated functions and provide analogs of some known non-periodic results. Under some additional assumptions error estimates are given in other terms in particular using the best approximation. A number of examples are provided.