This paper presents a fully-vectorial, three-dimensional algorithm for computing the definite-frequency eigenstates of Maxwell's equations in arbitrary periodic dielectric structures, including anisotropic and magnetic materials. The method uses preconditioned block-iterative eigensolvers in a planewave basis, achieving favorable scaling with system size and number of computed bands. A new effective dielectric tensor is introduced for anisotropic structures, enabling $ O(\Delta x^{2}) $ convergence even in systems with sharp material discontinuities. The algorithm can solve for interior eigenvalues, such as localized defect modes, without computing the many underlying eigenstates. Preconditioned conjugate-gradient Rayleigh-quotient minimization is compared with the Davidson method for eigensolution, and various iteration variants and preconditioners are characterized. The implementation is freely available online. The paper discusses the eigenproblem formulation of Maxwell's equations, the choice of basis functions, and the effective dielectric tensor. It describes the planewave basis and its advantages, including the ability to maintain transversality constraints. The effective dielectric tensor is introduced to handle discontinuities and anisotropy, with a smoothed effective tensor that improves convergence. The paper also addresses the choice of preconditioners, including diagonal and more accurate inverse preconditioners, and the removal of singularities at $ \vec{k}=0 $. Iterative eigensolvers are discussed, including the conjugate-gradient minimization of the Rayleigh quotient and the Davidson method. These methods are shown to be effective for computing the few lowest eigenstates of Maxwell's equations. The paper presents results for a 3D diamond lattice of dielectric spheres, demonstrating the convergence of the algorithms and the effectiveness of the preconditioners. The implementation is available as a free and flexible computer program.This paper presents a fully-vectorial, three-dimensional algorithm for computing the definite-frequency eigenstates of Maxwell's equations in arbitrary periodic dielectric structures, including anisotropic and magnetic materials. The method uses preconditioned block-iterative eigensolvers in a planewave basis, achieving favorable scaling with system size and number of computed bands. A new effective dielectric tensor is introduced for anisotropic structures, enabling $ O(\Delta x^{2}) $ convergence even in systems with sharp material discontinuities. The algorithm can solve for interior eigenvalues, such as localized defect modes, without computing the many underlying eigenstates. Preconditioned conjugate-gradient Rayleigh-quotient minimization is compared with the Davidson method for eigensolution, and various iteration variants and preconditioners are characterized. The implementation is freely available online. The paper discusses the eigenproblem formulation of Maxwell's equations, the choice of basis functions, and the effective dielectric tensor. It describes the planewave basis and its advantages, including the ability to maintain transversality constraints. The effective dielectric tensor is introduced to handle discontinuities and anisotropy, with a smoothed effective tensor that improves convergence. The paper also addresses the choice of preconditioners, including diagonal and more accurate inverse preconditioners, and the removal of singularities at $ \vec{k}=0 $. Iterative eigensolvers are discussed, including the conjugate-gradient minimization of the Rayleigh quotient and the Davidson method. These methods are shown to be effective for computing the few lowest eigenstates of Maxwell's equations. The paper presents results for a 3D diamond lattice of dielectric spheres, demonstrating the convergence of the algorithms and the effectiveness of the preconditioners. The implementation is available as a free and flexible computer program.