The authors report on the observation of enhanced microwave transmission through quasiperiodic hole arrays in metal films. The fraction of transmitted light reaches 50% in a self-standing metal film and approaches 90% when the film is sandwiched between thin dielectric slabs, while the holes occupy only 10% of the sample area. The maximum transmission exhibits a Breit-Wigner resonance behavior, accompanied by zero phase change and rendering the film almost invisible over a wide frequency range. The extraordinary transmission phenomenon is interpreted in terms of resonances in the self-consistent interaction between holes, which are represented by effective electric and magnetic dipoles.