Electronic equipment may include structured fabric. Structured fabric may be used as a protective case or cosmetic cover for an electronic device, may be used to form a band that holds an electronic device against a user's body, or may be used to cover one or more openings in an electronic device. Structured fabrics may be soft and pliable while maintaining the ability to hold a given shape without added support. Structured fabric may be formed by laminating fabric such as warp-knit fabric with a stiffener such as polymer film. Structured fabrics may include openings through which signals such as optical or audio signals pass. To maintain the geometry and shape of the openings in the structured fabric without covering the openings, the stiffener and adhesive that are attached to the fabric may be cut to form a pattern of openings that align with the openings in the fabric.