Kinetic roughening during homoepitaxial growth was studied for Ag(111) and Ag(001). For Ag(111), from 150 to 500 K, the rms roughness exhibits a power law, {sigma}{proportional_to}{ital t}{sup {beta}} over nearly three decades in thickness. {beta}{approx_equal}1/2 at low temperatures, and there is an abrupt transition to smaller values above 300 K. In contrast, Ag(001) exhibits layer-by-layer growth with a significantly smaller {beta}. These results are the first to establish the evolution of surface roughness quantitatively for a broad thickness and temperature range, as well as for the case where growth kinetics are dominated by a step-ledge diffusion barrier. {copyright} {ital 1996 The American Physical Society.}