  We present spectroscopic observations obtained with the infrared Spitzer
Space Telescope, which provide insight into the H$_2$ physics and gas
energetics in photodissociation Regions (PDRs) of low to moderate
far-ultraviolet (FUV) fields and densities. We analyze data on six well known
Galactic PDRs (L1721, California, N7023E, Horsehead, rho Oph, N2023N), sampling
a poorly explored range of excitation conditions ($\chi \sim 5-10^3$), relevant
to the bulk of molecular clouds in galaxies. Spitzer observations of H$_2$
rotational lines are complemented with H$_2$ data, including ro-vibrational
line measurements, obtained with ground-based telescopes and ISO, to constrain
the relative contributions of ultraviolet pumping and collisions to the H$_2$
excitation. The data analysis is supported by model calculations with the
Meudon PDR code. The observed column densities of rotationally excited H$_2$
are observed to be much higher than PDR model predictions. In the lowest
excitation PDRs, the discrepancy between the model and the data is about one
order of magnitude for rotational levels $J \ge $3. We discuss whether an
enhancement in the H$_2$ formation rate or a local increase in photoelectric
heating, as proposed for brighter PDRs in former ISO studies, may improve the
data-model comparison. We find that an enhancement in the H$_2$ formation rates
reduces the discrepancy, but the models still fall short of the data. This
large disagreement suggests that our understanding of the formation and
excitation of H$_2$ and/or of PDRs energetics is still incomplete. We discuss
several explanations, which could be further tested using the Herschel Space
Telescope
