We present 10, 20, and 30 microns maps of the central 60" x 90" (R.A. × decl.) of the Galaxy made at approximately 4" resolution with the NASA/Marshall Space Flight Center bolometer array. The maps span 2.5 × 3.8 pc centered near Sgr A IRS 1 and are the first to show the thermal emission from dust particles in both the ionized cavity and the neutral-gas ring with high sensitivity and an angular resolution as good as several arcseconds. In addition to warm dust associated with previously identified filaments in the central 40", these maps show the detailed distribution of dust along the entire Western Arc of ionized gas at the inner edge of the neutral-gas ring. A prominent tongue of high 30 microns optical depth extends from the northern part of the ring into the cavity near IRS 1, nearly bisecting the cavity; since we detect warm dust in this 30 mum-emitting feature, which we call the Northern Intruder, it must be heated by radiation emitted in the cavity, thus confirming previous speculations based on far-infrared and O0 observations that substantial neutral material protrudes into the cavity and may constitute infalling matter that fuels the central activity. We show that all the major ionized filaments (the Western Arc, the Bar, the Northern Arm, and the Eastern Arm) are ionization fronts at the interfaces between low- and high-density regions, as had been previously demonstrated convincingly only for the Western Arc. The locations of these ionization fronts are consistent with the dominant UV heating and ionizing sources being centrally located in the cavity. The derived dust temperatures strongly support this picture: they decrease away from the region of IRS 1 and Sgr A*, and they drop abruptly where the gas density increases at the ring. We determine that the total ultraviolet luminosity emitted by the central cluster and required to heat the dust to the inferred temperatures is 1.7±0.5 × 107 Lsun, which agrees with our derived infrared luminosity. Half of this ultraviolet luminosity is absorbed by dust in the cavity, and more than 40% of the ultraviolet radiation that propagates to the edge of the cavity is absorbed by dust in the neutral-gas ring.