Twisted tensor products include ordinary tensor products, semi-direct products with groups, crossed products with Hopf algebras, universal enveloping algebras, quantum polynomial rings, Weyl algebras, Ore extensions, Sridharan algebras, Koszul pairs, and many noncommutative structures in general. We give a method for building resolutions of twisted tensor products useful for computing various kinds of homology and cohomology. We recover different constructions in the literature as special cases, for example, the Chevalley-Eilenberg resolution for Lie algebra cohomology.