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Using terahertz time-domain spectroscopy, we have measured the optical properties of metallic films perforated with arrays of subwavelength apertures that possess short-range order (SRO), but lack long-range orientational order (LRO). We demonstrate that extraordinary transmission enhancement still occurs through the SRO aperture structures, despite the absence of LRO. The dielectric response of these arrays is characterized by a superposition of a broad principal resonance that is due to the random rotations of the building block (BB) units, and discrete resonances arising from well-defined reciprocal vectors in the structure factor that result from a virtual lattice associated with the BB units.