An analysis of motions consistent with uniform rotation in dense cores is presented. Twenty-nine of the 43 cores studied have a statistically significant gradient. Some gradients are spatially continuous and are consistent with uniform rotation, but other apparent gradients are caused by clump-clump motion, or sharp localized gradients, within a map. The motions in L1495, B217, L1251, L43, B361, and L1551 are discussed in detail. In L1551, the residuals of the fit to the NH3 velocity field indicate an outflow from IRS5 in the same direction as the CO outflow. Gradient orientation appears to be preserved over a range of density, as evidenced by comparing results of NH3 to fits of (C-18)O and CS maps. The specific angular momentum is found to scale roughly as F exp 3/2, where R represents the diameter of the FWHM intensity contour in a map.