[en] Checkerboard patterns have been proposed to explain the real space structure observed in scanning tunneling microscopy experiments on BSCCO and Na-CCOC. However, simple checkerboard patterns have low energy incommsensurate (IC) spin peaks rotated 45 deg. from the direction of the charge IC peaks, contrary to what is seen in neutron scattering. Here, we study modulated checkerboard patterns which can resolve the low frequency inconsistency. Using spin-wave theory, we explore the spin response of these superstructures and find that the high energy response is inconsistent with neutron scattering results. In particular, the modulated checkerboard structures are incapable of supporting the experimentally well-established resonance peak at (π,π)