We have studied the effect of thermal annealing on electron dephasing times τØ in three-dimensional polycrystalline metals. Measurements are performed on as-sputtered and annealed AuPd and Sb thick films, using the weak-localization method. In all samples, we find that τØ possesses an extremely weak temperature dependence as T → 0. Our results show that the effect of annealing is non-universal, and it depends strongly on the amount of disorder quenched in the microstructures during deposition. The observed "saturation" behavior of τØ cannot be easily explained by magnetic scattering. We suggest that the issue of saturation can be better addressed in three-dimensional, rather than lower-dimensional, structures.