Let G be a finite group and let R be a commutative ring with unit. One of the most basic problems in group cohomology is that of finding suitable projective resolutions for RG-modules. The main result of this paper shows that if \(R=k\) is a field of characteristic \(p>0\), and if M is any finitely generated kG-module then there exists a projective resolution of M that is a tensor product of periodic complexes. The constructed resolutions are usually not minimal but do have the same rate of growth (complexity) as the corresponding minimal resolutions. An integral version of the main theorem for RG-lattices is also proved. As an application of the results we present a new proof of G. Carlsson's theorem that a finite group acting freely on a product of n spheres of the same dimension must have p-rank at most n for all primes p.