The dielectric properties of ceramics pressurelessly sintered from nano (10-100 nm) and fine (0.8-1.0 mum) aluminum nitride powders have been studied. High-purity aluminum nitride sintered from nanopowder of nearly-theoretical density has been found to exhibit a low dielectric constant epsilon = 63 and low loss, tgdelta = 0.011 at 3.32 GHz. Ceramics sintered from a fine aluminum nitride powder doped with yttrium oxide have revealed epsilon = 7.7 and tgdelta = 0.021 at 3.23 GHz. The dielectric properties have allowed us to use ceramics sintered from fine aluminum nitride powder as a material for microwave windows for the diagnosis and treatment of tumors of the larynx and breast.