Multi-layered golf ball core sub-assemblies and the resulting golf balls are provided. The core structure includes an inner core (center) comprising a foam composition, preferably foamed polyurethane. The intermediate core layer is preferably formed from a thermoplastic composition such as an ethylene acid copolymer ionomer resin; and the outer core layer is preferably formed from a thermoset composition such as polybutadiene rubber. The core layers have different hardness and specific gravity levels. In one preferred embodiment, the inner core has a positive hardness gradient, the intermediate core has a positive hardness gradient, and the outer core has a zero or negative hardness gradient. The core structure and resulting ball have relatively good resiliency.