Machining additively manufactured parts to improve surface finish is, sometimes, a challenging process because different manufacturing orientations and process parameters result in different machined surfaces. Extensive research has been devoted to additive manufacturing (AM) technologies and their abilities to produce end-use parts. However, post processing such as machining is still required for many applications. In this paper, non-traditional machining, Wire Electric Discharge Machining (WEDM), is used to study the result of WEDM on the surface roughness and machining time when the layer thickness and cutting orientation are changed. In this study, three layers’ thicknesses (LT = 60, 80, 100 μm) and three orientations of cut are considered; wire cuts in a perpendicular plane to a layer (Ori-1), wire cuts in a parallel plane to a layer (Ori-2), and wire cuts all layers at the same time (Ori-3). Surface profilometry, scanning electron microscope (SEM), and machining time are used to evaluate the machining performance. The results show that the variation of the machining time for all layer thicknesses on Ori-1 and Ori-2 was minimal, except for Ori-3. Similarly, better surface quality could be achieved when the WEDM cuts along the Ori-2, whereas the variation of surface roughness (S_a) increases for the Ori-3. LT is also found to affect machining performance. For instance, when LT = 100 μm is employed, the roughness variation among different orientations increases up to 22.36%. On the other hand, roughness variation among different orientations reduced to 2.42% and 10.7% for LT of 60 μm and 80 μm, respectively. Overall, it was found that the WEDM along the Ori-2 yields improved surface roughness for any layer thickness. Similarly, the SEM results also show that improved surface morphology with less redeposited material could be achieved when the parts are machined along the Ori-2 as compared to Ori-1 and Ori-3.