Resonantly enhanced light transmission through periodic subwavelength aperture arrays perforated in metallic films has generated significant interest because of potential applications in near-field microscopy, photolithography, displays, and thermal emission. The enhanced transmission was originally explained by a mechanism where surface plasmon polaritons (collective electronic excitations in the metal surface) mediate light transmission through the grating. In this picture, structural periodicity is perceived to be crucial in forming the transmission resonances. Here we demonstrate experimentally that, in contrast to the conventional view, sharp transmission resonances can be obtained from aperiodic aperture arrays. Terahertz transmission resonances are observed from several arrays in metallic films that exhibit unusual local n-fold rotational symmetries, where n = 10, 12, 18, 40 and 120. This is accomplished by using quasicrystals with long-range order, as well as a new type of 'quasicrystal approximates' in which the long-range order is somewhat relaxed. We find that strong transmission resonances also form in these aperiodic structures, at frequencies that closely match the discrete Fourier transform vectors in the aperture array structure factor. The shape of these resonances arises from Fano interference of the discrete resonances and the non-resonant transmission band continuum related to the individual holes. Our approach expands potential design parameters for aperture arrays that are aperiodic but contain discrete Fourier transform vectors, and opens new avenues for optoelectronic devices.