A search has been made for the simplest amino acid - glycine - in Sgr B2, Ori A, and seven other molecular clouds. Six different lines were sought ranging from 83.4 GHz down to 22.7 GHz, and radio telescopes at Parkes, New South Wales, Kitt Peak, Arizona, and Onsala, Sweden, were used. On the assumption of reasonable values for excitation temperature and Boltzmann distributions over rotational energy states, upper limits of column densities for glycine of a few times 10 to the 12th per sq cm to a few times 10 to the 14th per sq cm were typically established in the sources surveyed. Glycine appears to be less abundant than ethanol or ethyl cyanide in molecular clouds such as Sgr B2 and Ori A.