The paper is devoted to the analysis of the stress-strain state of an elastic isotropic layer weakened by a system of three parallel circular cylindrical cavities under spatially periodic loading. The presence of internal cavities acts as a stress concentrator, which significantly complicates the analysis of structural elements operating under periodic loads. To solve the spatial problem of elasticity theory, a generalized Fourier method is applied. The solution to the problem is presented as a superposition of two solutions: the auxiliary and the main problem. The method uses a combination of different coordinate systems: Cartesian for describing the layer and local cylindrical for each of the cavities. Satisfying the boundary conditions on the surfaces of the layer and cavities leads to an infinite system of linear algebraic equations with respect to the unknown decomposition coefficients. For numerical implementation, the system is truncated to a finite order. Within the framework of the work, a numerical analysis was performed for a layer with three cavities of the same radius. An anisotropic reaction of the material was revealed, caused by the specific location of the cavities along one axis. It was established that an external spatially periodic compressive load leads to the emergence of a complex stress field. The research results are of practical importance for the design and optimization of new composite materials, as well as for refining strength calculations for elements operating under periodic loads.