We study coherent phonon transport through organic, \pi-conjugated molecules. Using first principles calculations and Green's function methods, we find that the phonon transmission function in cross-conjugated molecules, like meta-connected benzene, exhibits destructive quantum interference features very analogous to those observed theoretically and experimentally for electron transport in similar molecules. The destructive interference features observed in four different cross-conjugated

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... les significantly reduce the thermal conductance with respect to linear conjugated analogues. Such control of the thermal conductance by chemical modifications could be important for thermoelectric applications of molecular junctions.