The ternary microemulsion systems alkane/water/DDAB (didodecyldimethylammonium bromide) form ideal model-disordered media. The static microstructure is described by a simple parameter-free model that can be predetermined and agrees with SAYS and SANS scattering experiments. The component volume fraction can be varied to exhibit bicontinuous random structures with a predicted percolation transition to disconnected water-in-oil droplets. Structural transitions are analyzed in the context of theories of percolative phenomena. Experimental transport properties agree well with model predictions based on an effective medium approximation. Critical exponents that describe the scaling of the transport properties near percolation are consistent with theoretical expectations near a static percolation transition. Through variation of component volume fractions a medium of known microstructure can be prescribed, so that independent measurement of transport and mechanical properties is possible.