We present an analysis of the radio properties of large samples of Lyman break galaxies (LBGs) at z ~ 3, 4, and 5 from the COSMOS field. The median stacking analysis yields a statistical detection of the z ~ 3 LBGs (U-band dropouts), with a 1.4 GHz flux density of 0.90 ± 0.21 μJy. The stacked emission is unresolved, with a size <1, or a physical size <8 kpc. The total star formation rate implied by this radio luminosity is 31 ± 7 M yr−1, based on the radio-FIR correlation in low-redshift star-forming galaxies. The star formation rate derived from a similar analysis of the UV luminosities is 17 M yr−1, without any correction for UV dust attenuation. The simplest conclusion is that the dust attenuation factor is 1.8 at UV wavelengths. However, this factor is considerably smaller than the standard attenuation factor of ~5, normally assumed for LBGs. We discuss potential reasons for this discrepancy, including the possibility that the dust attenuation factor at z ≥ 3 is smaller than at lower redshifts. Conversely, the radio luminosity for a given star formation rate may be systematically lower at very high redshift. Two possible causes for a suppressed radio luminosity are (1) increased inverse Compton cooling of the relativistic electron population due to scattering off the increasing CMB at high redshift or (2) cosmic-ray diffusion from systematically smaller galaxies. The radio detections of individual sources are consistent with a radio-loud AGN fraction of 0.3%. One source is identified as a very dusty, extreme starburst galaxy (a "submillimeter galaxy").