Motivated by recent progress in epitaxial growth of proximity structures of s-wave superconductors (S) and spin-active materials (M), in this paper we show that certain periodic structures of S and M can behave effectively as superconductors with pairs of point nodes, near which the low-energy excitations are Weyl fermions. A simple model, where M is described by a Kronig-Penney potential with both spin-orbit coupling and exchange field, is proposed and solved to obtain the phase diagram of the nodal structure, the spin texture of the Weyl fermions, as well as the zero-energy surface states in the form of open Fermi lines (Fermi arcs). As a second example, a lattice model with alternating layers of S and magnetic Z(2) topological insulators is solved. The calculated spectrum confirms previous predictions of Weyl nodes based on the tunnelling Hamiltonian of Dirac electrons. Our results provide further evidence that periodic structures of S and M are well suited for engineering gapless topological superconductors.