We have modeled the physical and chemical state of dense (n = 10(3)-10(5) cm(-3)) neutral gas exposed to intense X-ray fluxes and the resultant infrared and submillimeter emission from the irradiated gas. The dominant parameter controlling the state of the gas in these X-ray dissociation regions (XDRs) is the ratio of local X-ray energy deposition rate to gas density; this can be expressed in terms of an effective ionization parameter. The parameter space we model ranges over 5 orders of magnitude in this ionization parameter; the gas physical conditions vary from warm, atomic, and partially ionized (T similar to 10(4) K, molecular hydrogen abundance x(H2) less than or similar to 10(-4), electron abundance x(e) similar to 0.1) to cold, molecular, and neutral (T similar to 20 K, x(H2) approximate to 0.5, x(e) similar to 10(-6)). We thus cover the entire range of parameter space in which X-ray ionization and heating is important and the gas is largely neutral. Although we assume a power law for the incident X-ray flux, most of our results are independent of this assumption and are of general applicability. A wide range of diagnostic atomic and molecular line emission is produced by XDRs, which are luminous sources of infrared and submillimeter lines. This is a consequence of the large column densities (N > 10(22) Cm-2) that hard (E > 1 keV photons are capable of penetrating before being absorbed. Strong emission lines include [Fe II] 1.26 and 1.64 mu m, [O I] 63 mu m, [C II] 158 mu m, [Si II] 35 mu m, and the 2 mu m vibration-rotation lines of H-2, such as the v = 1-0 S(1) line at 2.12 mu m. We discuss diagnostic line ratios for discriminating XDRs from shocks and photodissociation regions. One strong signature of emission from XDRs is the large flux ratio (similar to 0.1) of the major coolant line fluxes (e.g., [O I] 63 mu m) to the bolometric continuum flux. XDRs are likely to be the dominant sources of emission in a range of astrophysical environments, such as molecular clouds within roughly 1 kpc of typical active galactic nuclei.