Inelastic neutron scattering (INS) from polycrystalline antiferromagnetic LaMnAsO, LaMnSbO, and BaMnAsF are analyzed using a J(1)-J(2)-J(c) Heisenberg model in the framework of the linear spin-wave theory. All three systems show clear evidence that the nearest- and next-nearest-neighbor interactions within the Mn square lattice layer (J(1) and J(2)) are both antiferromagnetic (AFM). However, for all compounds studied the competing interactions have a ratio of 2J(2)/J(1) < 1, which favors the square lattice checkerboard AFM structure over the stripe AFM structure. The interplane coupling Jc in all three systems is on the order of similar to 3 x 10(-4)J(1), rendering the magnetic properties of these systems with quasi-two-dimensional character. The substitution of Sb for As significantly lowers the in-plane exchange coupling, which is also reflected in the decrease of the Neel temperature, T-N. Although BaMnAsF shares the same MnAs sheets as LaMnAsO, their J(1) and J(2) values are substantially different. Using density functional theory, we calculate exchange parameters J(ij) to rationalize the differences among these systems.