We have measured the {sup 17}O nuclear-magnetic-resonance spin-lattice relaxation times ({ital T}{sub 1}) as a function of temperature for the CuO{sub 2} planar sites in Tl{sub 2}Ba{sub 2}CaCu{sub 2}O{sub 8+{ital x}}, YBa{sub 2}Cu{sub 3}O{sub 7{minus}{ital x}}, Bi{sub 2}Sr{sub 2}CaCu{sub 2}O{sub 8+{ital x}}, La{sub 1.85}Sr{sub 0.15}CuO{sub 4{minus}{ital x}}, La{sub 1.85}Ca{sub 0.15}CuO{sub 4{minus}{ital x}}, and Bi{sub 2}Sr{sub 2}CuO{sub 6+{ital x}}, as well as for the oxygen sites in Ba{sub 0.6}K{sub 0.4}BiO{sub 3}, BaBi{sub 0.25}Pb{sub 0.75}O{sub 3}, BaSb{sub 0.25}Pb{sub 0.75}O{sub 3}, and BaPbO{sub 3}. The CuO{sub 2} planar sites in Tl{sub 2}Ba{sub 2}CaCu{sub 2}O{sub 8+{ital x}}, YBa{sub 2}Cu{sub 3}O{sub 7{minus}{ital x}}, and Bi{sub 2}Sr{sub 2}CaCu{sub 2}O{sub 8+{ital x}} exhibit close to ideal Korringa behavior in the normal state, characteristic of a good metal. In addition, the Knight shift of the CuO{sub 2} planes in these three materials is dominated by a spin contribution that is temperature independent in the normal state and vanishes in the superconducting state. The relaxation times of the BaO and SrO planar oxygens, as well as of the TlO planar oxygens, are much longer than those of the CuO{sub 2} planes, and are similar to the values found for the bismuthate and plumbate materials. In both lanthanum cuprates, themore » frequency shift of the CuO{sub 2} plane is temperature dependent in the normal state. Relaxation-rate data, when plotted as a function of ln({ital T}{sub {ital c}}), show a clear difference between the bismuthate (plumbate) and cuprate materials, and appear consistent with BCS-like behavior for the copper-free systems.« less