Epitaxial strain imposed in complex oxide thin films by heteroepitaxy is recognized as a powerful tool for identifying new properties and exploring the vast potential of materials performance. A particular example is LaCoO(3), a zero spin, nonmagnetic material in the bulk, whose strong ferromagnetism in a thin film remains enigmatic despite a decade of intense research. Here, we use scanning transmission electron microscopy complemented by X-ray and optical spectroscopy to study LaCoO(3) epitaxial thin films under different strain states. We observed an unconventional strain relaxation behavior resulting in stripe-like, lattice modulated patterns, which did not involve uncontrolled misfit dislocations or other defects. The modulation entails the formation of ferromagnetically ordered sheets comprising intermediate or high spin Co(3+), thus offering an unambiguous description for the exotic magnetism found in epitaxially strained LaCoO(3) films. This observation provides a novel route to tailoring the electronic and magnetic properties of functional oxide heterostructures.