We have measured the production of both Lyα and Hα fluorescence from atomic H and D for the photodissociation of H2 and D2 by linearly polarized photons with energies between 24 and 60 eV. In this energy range, excited photofragments result primarily from the production of doubly excited molecular species which promptly autoionize or dissociate into two neutrals. Our data are compared with ab initio calculations of the dissociation process, in which both doubly excited state production and prompt ionization (non-resonant) channels are considered. Agreement between our experimental data and that of earlier work, and with our theoretical calculations, is qualitative at best.