Multi-piece golf balls containing a dual-core structure are provided. The core structure includes an inner core (center) comprising a foam composition, preferably foamed polyurethane. The outer core layer is preferably formed from a non-foamed thermoset composition such as polybutadiene rubber. The inner and outer core layers can have different hardness gradients. Preferably, the specific gravity (density) of the foam inner core is less than the density of the outer core layer. The ball further includes a cover of at least one layer and may include at least one casing layer. The core structure and resulting ball have relatively good resiliency.