The skins of many plants and animals have intricate microscale surface features that give rise to properties such as directed water repellency and adhesion, camouflage, and resistance to fouling. However, engineered mimicry of these designs has been restrained by the limited capabilities of top-down fabrication processes. Here we demonstrate a new technique for scalable manufacturing of freeform microstructures via strain-engineered growth of aligned carbon nanotubes (CNTs). Offset patterning of the CNT growth catalyst is used to locally modulate the CNT growth rate. This causes the CNTs to collectively bend during growth, with exceptional uniformity over large areas. The final shape of the curved CNT microstructures can be designed via finite element modeling, and compound catalyst shapes produce microstructures with multidirectional curvature and unusual self-organized patterns. Conformal coating of the CNTs enables tuning of the mechanical properties independently from the microstructure geometry, representing a versatile principle for design and manufacturing of complex microstructured surfaces.