We examine the effective diffusion coefficient of composites containing randomly placed, impermeable, oriented disks of high aspect ratio. We extend previous work by providing a quantitative framework in which to examine disorder and polydispersity effects, as well as treating explicitly the crossover from the dilute to the semidilute regime as the concentration of disks is increased. Our results have applicability to matrices filled with macroscopic plates or flakes for example, mica-filled polymers) as well as to polymer-clay hybrid materials ("nanocomposites＇＇).