The oxychalcogenides A(2)F(2)Fe(2)OQ(2) (A = Sr, Ba; Q = S, Se), which contain Fe2O square planar layers of the anti-CuO2 type, were predicted using a modular assembly of layered secondary building units and subsequently synthesized. The physical properties of these compounds were characterized using magnetic susceptibility, electrical resistivity, specific heat, (57) Fe Mossbauer, and powder neutron diffraction measurements and also by estimating their exchange interactions on the basis of first-principies density functional theory electronic structure calculations. These compounds are magnetic semiconductors that undergo a long-range antiferromagnetic ordering below 83.6-106.2 K, and their magnetic properties are well-described by a two-dimensional Ising model. The dominant antiferromagnetic spin exchange interaction between S = 2 Fe2+ ions occurs through corner-sharing Fe-O-Fe bridges. Moreover, the calculated spin exchange interactions show that the A(2)F(2)Fe(2)OQ(2) (A = Sr, Ba; Q = S, Se) compounds represent a rare example of a frustrated antiferromagnetic checkerboard lattice.