The paper presents a fully-vectorial, three-dimensional algorithm for computing 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, demonstrating favorable scaling with system size and the number of computed bands. A new effective dielectric tensor is introduced for anisotropic structures, achieving $O(\Delta a^2)$ convergence even in systems with sharp material discontinuities. The paper also describes techniques for solving interior eigenvalues, such as localized defect modes, without computing underlying bands. Comparisons are made between different iterative solution schemes, preconditioners, and aspects of frequency-domain calculations. The implementation is freely available online. The authors discuss the choice of basis functions, the effective dielectric tensor, and various block-iterative algorithms for solving the eigensystem, emphasizing the importance of transversality constraints and the benefits of using a planewave basis. They also explore the impact of inversion symmetry and propose a method for defining the surface normal at dielectric interfaces. The paper concludes with a detailed comparison of the Davidson method and the block conjugate-gradient algorithm, highlighting the advantages of the former in terms of convergence and computational efficiency.The paper presents a fully-vectorial, three-dimensional algorithm for computing 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, demonstrating favorable scaling with system size and the number of computed bands. A new effective dielectric tensor is introduced for anisotropic structures, achieving $O(\Delta a^2)$ convergence even in systems with sharp material discontinuities. The paper also describes techniques for solving interior eigenvalues, such as localized defect modes, without computing underlying bands. Comparisons are made between different iterative solution schemes, preconditioners, and aspects of frequency-domain calculations. The implementation is freely available online. The authors discuss the choice of basis functions, the effective dielectric tensor, and various block-iterative algorithms for solving the eigensystem, emphasizing the importance of transversality constraints and the benefits of using a planewave basis. They also explore the impact of inversion symmetry and propose a method for defining the surface normal at dielectric interfaces. The paper concludes with a detailed comparison of the Davidson method and the block conjugate-gradient algorithm, highlighting the advantages of the former in terms of convergence and computational efficiency.