Corrugated furring strips are used on the outer side of an exterior insulation layer in an exterior wall construction. The corrugated furring strips support a final cladding layer at an offstanding position from the insulation, thereby providing a rainscreen. The corrugated furring strips are fastened to the framework of the wall through the exterior insulation layer. Optional bridging members shaped to conformingly mate with lands and grooves of the corrugated furring straps cooperate therewith to define a support grid with improved load capacity for heavier cladding materials and more fastening location options. The insulation layer is free of any thermal bridges other than the insulation-penetrating fasteners used to support the furring and optional bridges. Thermally insulative washers isolate the furring and optional bridges from the fasteners to further improve the insulating effect of the wall.