One illustrative method disclosed herein includes, among other things, sequentially forming a first material layer, a first capping layer, a second material layer and a second capping layer above a substrate, wherein the first and second material layers are made of semiconductor material having a lattice constant that is different than the substrate, the first material layer is strained as deposited, and a thickness of the first material layer exceeds its critical thickness required to be stable and strained, performing an anneal process after which the strain in the first material layer is substantially relaxed through the formation of crystallographic defects that are substantially confined to the semiconducting substrate, the first material layer, the first capping layer and the second material layer, and forming additional epitaxial semiconductor material on an upper surface of the resulting structure.