A method of laser welding a workpiece stack-up that includes two or three aluminum alloy workpieces (14, 16), the method comprising: providing a workpiece stack-up that includes at least a first aluminum alloy workpiece and an underlying second aluminum alloy workpiece that overlap and contact one another to establish a first faying interface between the workpieces (14, 16), the first aluminum alloy workpiece providing a top surface of the workpiece stack-up; constraining a free end of an overlapping portion of the first aluminum alloy workpiece against movement away from the underlying second aluminum alloy workpiece, wherein constraining the free end of the overlapping portion of the first aluminum alloy workpiece comprises clamping the free end against the second aluminum alloy workpiece; directing a laser beam at the top surface of the workpiece stack-up to create a keyhole within the workpiece stack-up and a molten aluminum alloy weld pool that surrounds the keyhole, the molten aluminum alloy weld pool traversing at least the first faying interface established between the first and second aluminum alloy workpieces (14, 16); and conveying the laser beam and the associated keyhole and molten aluminum alloy weld pool relative to the top surface of the workpiece stack-up along a travel path within a welding region, the welding region being located within the overlapping portion of the first aluminum alloy workpiece inward of the free end; characterized in that disposing a clamping support shim adjacent to the free end of the overlapping portion of the first aluminum alloy workpiece over the second aluminum alloy workpiece, wherein the clamping support shim has a height that corresponds to the height of the free end of the first aluminum alloy workpiece and has an upper surface; engaging one or more clamps against both the top surface of the workpiece stack-up and the upper surface of the clamping support shim, thereby directing a downward force against both the top surface of the workpiece stack-up and the upper surface of the clamping support shim in order to clamp the free end of the first aluminum alloy workpiece against the second aluminum alloy workpiece. providing a workpiece stack-up that includes at least a first aluminum alloy workpiece and an underlying second aluminum alloy workpiece that overlap and contact one another to establish a first faying interface between the workpieces (14, 16), the first aluminum alloy workpiece providing a top surface of the workpiece stack-up; constraining a free end of an overlapping portion of the first aluminum alloy workpiece against movement away from the underlying second aluminum alloy workpiece, wherein constraining the free end of the overlapping portion of the first aluminum alloy workpiece comprises clamping the free end against the second aluminum alloy workpiece; directing a laser beam at the top surface of the workpiece stack-up to create a keyhole within the workpiece stack-up and a molten aluminum alloy weld pool that surrounds the keyhole, the molten aluminum alloy weld pool traversing at least the first faying interface established between the first and second aluminum alloy workpieces (14, 16); and conveying the laser beam and the associated keyhole and molten aluminum alloy weld pool relative to the top surface of the workpiece stack-up along a travel path within a welding region, the welding region being located within the overlapping portion of the first aluminum alloy workpiece inward of the free end; characterized in that disposing a clamping support shim adjacent to the free end of the overlapping portion of the first aluminum alloy workpiece over the second aluminum alloy workpiece, wherein the clamping support shim has a height that corresponds to the height of the free end of the first aluminum alloy workpiece and has an upper surface; engaging one or more clamps against both the top surface of the workpiece stack-up and the upper surface of the clamping support shim, thereby directing a downward force against both the top surface of the workpiece stack-up and the upper surface of the clamping support shim in order to clamp the free end of the first aluminum alloy workpiece against the second aluminum alloy workpiece.