  Motivated by ongoing measurements at JILA, we calculate the recoil-free
spectra of dipolar interacting fermions, for example ultracold heteronuclear
molecules, in a one-dimensional lattice of two-dimensional pancakes,
spectroscopically probing transitions between different internal (e.g.,
rotational) states. We additionally incorporate p-wave interactions and losses,
which are important for reactive molecules such as KRb. Moreover, we consider
other sources of spectral broadening: interaction-induced quasiparticle
lifetimes and the different polarizabilities of the different rotational states
used for the spectroscopy. Although our main focus is molecules, some of the
calculations are also useful for optical lattice atomic clocks. For example,
understanding the p-wave shifts between identical fermions and small dipolar
interactions coming from the excited clock state are necessary to reach future
precision goals. Finally, we consider the spectra in a deep 3D lattice and show
how they give a great deal of information about static correlation functions,
including \textit{all} the moments of the density correlations between nearby
sites. The range of correlations measurable depends on spectroscopic resolution
and the dipole moment.
