By combining a state-of-the-art high-harmonic ultrafast soft X-ray source with field-free dynamic alignment, we map the angular dependence of molecular photoionization yields for the first time for a nondissociative molecule. The observed modulation in ion yield as a function of molecular alignment is attributed to the molecular frame transition dipole moment of single-photon ionization to the X. A and B states of N-2(+) and CO2+. Our data show that the transition dipoles for single-photon ionization of N-2 and CO2 at 43 eV have larger perpendicular components than parallel ones. A direct comparison with published theoretical partial wave ionization cross-sections confirms these experimental observations, which are the first results to allow such comparison with theory for bound cation states. The results provide the first step toward a novel method for measuring molecular frame transition dipole matrix elements.