We have used laser-micromachined alumina substrates to build a three-dimensional photonic band-gap crystal. The rod-based structure has a three-dimensional full photonic band gap between 90 and 100 GHz. The high resistivity of alumina results in a typical attenuation rate of 15 dB per unit cell within the band gap. By removing material, we have built defects which can be used as millimeter-wave cavity structures. The resulting quality ~Q! factors of the millimeter-wave cavity structures were as high as 1000 with a peak transmission of 10 dB below the incident signal. © 1995 American Institute of Physics.