We report on the optical response and, in particular, on the refractive properties of an fcc crystal of metallic nanoshells by means of full-electrodynamic layer-multiple-scattering simulations. Exact numerical calculations of the isofrequency surfaces reveal the existence of two frequency regions where negative refraction occurs. A thorough analysis of the photonic band structure, in conjunction with corresponding transmission diagrams, attributes this behavior to the excitation of collective modes, which stem from dipole particle-plasmon resonances, and shows that only in one of the two frequency regions negative refraction without birefringence can be obtained. In addition, we discuss the effect of absorptive losses, and reveal the existence of narrow bands of slab modes in a finite slab of the crystal that can transfer the evanescent components of an incident wave field.