We introduce and optimize a fabrication procedure that employs both femtosecond laser machining and hydrofluoric acid etching for cutting holes or voids in slabs of lithium niobate and lithium tantalate. The fabricated structures have 3 μm lateral resolution, a lateral extent of at least several millimeters, and cut depths of up to 100 μm. Excellent surface quality is achieved by initially protecting the optical surface with a sacrificial silicon dioxide layer that is later removed during chemical etching. To optimize cut quality and machining speed, we explored various laser machining parameters, including laser polarization, repetition rate, pulse duration, pulse energy, exposure time, and focusing, as well as scanning, protective coating, and etching procedures. The resulting structures significantly broaden the capabilities of terahertz polaritonics, in which lithium niobate and lithium tantalate are used for terahertz wave generation, imaging, and control. The approach should be applicable to a wide range of materials that are difficult to process by conventional methods. Keywords femtosecond laser machining • hydrofluoric acid etch • lithium niobate • lithium tantalate • terahertz polaritonics • surface treatment • silicon dioxide • materials processing