A structure for use in industrial fabrics such as paper machine clothing and engineered fabrics is disclosed. The structure includes one or more layers of an elastic nonwoven extruded film or sheet, which is elastic, resilient, and compressible in a thickness direction, and extensible, bendable, and resilient in its length and transverse directions, and one or more layers of a plurality of substantially parallel machine direction (MD) yarns in various patterns. The structure can also include one or more layers of a plurality of substantially parallel cross-machine direction (CD) yarns attached on top of or under the MD yarns. The structure has a high degree of both compressibility under an applied normal load and excellent recovery (resiliency or spring back) upon removal of that load.