This paper presents a new Green's operator (Gamma-operator) for mixed periodic and stress-free boundary conditions. The solution is split into a classical periodic solution and a complementary solution to account for stress-free boundary conditions. This operator is used to obtain the effective properties of thin heterogeneous plates made of periodic cells, using the Fast Fourier Transform, the periodic thin plate theory and the computation of the strain energy within the cell. Numerical applications are provided, including comparisons with closed-form solutions and finite element solutions.