Recent research on the additive manufacturing (AM) of Ti alloys has shown that the mechanical properties of the parts are affected by the characteristic microstructure that originates from the AM process. To understand the effect of the microstructure on the tensile properties, selective laser melted (SLM) Ti–6Al–4V samples built in three different orientations were tensile tested. The investigated samples were near fully dense, in two distinct conditions, as-built and stress relieved. It was found that the build orientation affects the tensile properties, and in particular the ductility of the samples. The mechanical anisotropy of the parts was discussed in relation to the crystallographic texture, phase composition and the predominant fracture mechanisms. Fractography and electron backscatter diffraction (EBSD) results indicate that the predominant fracture mechanism is intergranular fracture present along the grain boundaries and thus provide and explain the typical fracture surface features observed in fracture AM Ti–6Al–4V.