We determine the characteristics of the 7 mm to 20 cm wavelength radio variability in Sgr A* on time scales from days to three decades. The amplitude of the intensity modulation is between 30 and 39% at all wavelengths. Analysis of uniformly sampled data with proper accounting of the sampling errors associated with the lightcurves shows that Sgr A* exhibits no 57- or 106-day quasi-periodic oscillations, contrary to previous claims. The cause of the variability is investigated by examining a number of plausible scintillation models, enabling those variations which could be attributed to interstellar scintillation to be isolated from those that must be intrinsic to the source. Thin-screen scattering models do not account for the variability amplitude on most time scales. However, models in which the scattering region is extended out to a radius of 50-500 pc from the Galactic Center account well for the broad characteristics of the variability on >4-day time scales. The 10% variability on <4-day time scales at 0.7-3 cm appears to be intrinsic to the source. The degree of scintillation variability expected at millimeter wavelengths depends sensitively on the intrinsic source size; the variations, if due to scintillation, would require an intrinsic source size smaller than that expected.