By means of synchrotron-radiation-based core-level spectroscopies we demonstrate that the degree of corrugation in graphene nanomesh on lattice-mismatched transition-metal substrates critically depends on the strength of chemical bonding at the interface. The degree of interfacial orbital hybridization between graphene and metal states is rising in the series Pt(111)-Ir(111)-Rh(111)-Ru(0001). This growing strength of hybridization is accompanied by a gradual change in graphene morphology from nearly flat to strongly corrugated. We provide a comparison of the pore size and period for the cases of graphene and h-BN nanomesh on Rh(111).