The photonic bandgap of three-dimensional photonic crystals, formed by arranging circular spirals in face-centre-cubic lattice, was theoretically investigated. The structure was found to have a relative photonic bandgap of up to 25% in both direct and inversed configurations. The conditions under which the structure has a bandgap larger than 10% are described. Some considerations for optimizing such photonic crystal fabrication by two-photon polymerization are given. The theoretical results are implemented to fabricate polymeric structures that can be used as templates for photonic crystals with full photonic bandgap larger than 10% centered in the near-infrared region.