We show that strong non-Franck-Condon effects can arise from photoionization of molecular Rydberg orbitals which evolve rapidly with internuclear distance and display Cooper minima in their photoelectron spectrum. The non-Franck-Condon mechanism should be most pronounced in excited-state photoionization of diatomic hydrides. The mechanism is illustrated with specific ab initio predictions of ion vibrational distributions for a proposed (3 + 1) one-color or (2 + 1) two-color resonant multiphoton ionization measurement of OH via the $D^{2}\ensuremath{\Sigma}^{\ensuremath{-}}(1{\ensuremath{\pi}}^{2}5\ensuremath{\sigma})$ state.