A method of laser welding a workpiece stack-up (10) that includes at least two overlapping aluminum workpieces (12, 14), at least one of which includes a protective anti-corrosion coating (38), is disclosed. The disclosed method includes advancing the laser beam (56) relative to the top surface (26) of the workpiece stack-up (10) along a travel path (78, 78′, 78″, 78′″) that imposes bidirectional movement of the laser beam (56). In particular, the laser beam (56) moves in a forward direction (80) while also moving back and forth in a lateral direction (82) oriented transverse to the forward direction (80) as it is being advanced relative to the top surface (26). Such bidirectional movement is believed to help disturb the protective anti-corrosion coating (38) in and around the molten aluminum weld pool (74), thus leading to a laser weld joint (68) that contains less weld defects derivable from the protective anti-corrosion coating(s) (38).