The failure of inhomogeneous microstructures is of increasing relevance, driven by the future need for tailored materials and the significant influence of microstructure on macroscopic properties. The phase-field method for fracture has proven to be a versatile tool for predicting unknown crack paths and failure mechanisms. In this contribution, we propose a phase-field model for fracture of periodic heterogeneous microstructures in a general finite strain setting. To overcome the bottleneck of scalability, we employ powerful and scalable solvers based on the fast Fourier transform (FFT). We demonstrate the capability of the model using the fundamental example of brittle fracture. A thorough comparison with conventional finite element method (FEM) reference results is carried out using a simple reference geometry. The results obtained show quantitative agreement between both numerical methods. Parameter studies provide recommendations for the choice of the numerical parameters. Following the comparison, we apply the FFT-based method to synthetic inhomogeneous microstructures, with a special emphasis of the investigation on scalability with increasing degrees of freedom and robustness of the method. The results of 2D and 3D simulations are promising, paving the way for future extensions in inverse materials design, for example, for metallic microstructures.