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.