Gravity and flow operated diverter valves are disclosed for diverting liquid flow from one source to either of two output destinations, the valves including a housing defining an inlet opening through a base wall, a first outlet opening through a first upper wall and a second outlet opening through a second upper wall. A valve member is movably suspended within the housing by a first end when no liquid is flowing into the inlet opening. The valve member is pivotable in a first direction to seal the first outlet opening and pivotable in a second direction to seal the second outlet opening. On successive liquid flows into the inlet opening the valve member automatically and alternatingly seals the first outlet opening and the second outlet opening. Related dishwasher designs are also disclosed.