In dimensions up to three, every manifold is triangulable (this is classical). In dimension 4, there are simply connected non-triangulable manifolds (such as the E8 manifold); in fact, a closed 4-manifold is triangulable if and only if it's smoothable. (Pick a triangulation; the links of vertices are always both homology and homotopy spheres, thus by the Poincare conjecture are $S^3$, so this gives a PL structure, and thus is smoothable. This is probably true more generally than closed manifolds, but I'm feeling a little paranoid.)
The crucial papers pre-Manolescu are, firstly, Galewski-Stern and Matumoto's thesis (which I don't have a link to): these prove that every closed manifold if dimension $n \geq 5$ are triangulable if and only if there is a homology 3-sphere $Y$ such that $Y \# Y$ bounds a homology ball, and the Rokhlin invariant $\mu(Y) = 1 \in \Bbb Z/2\Bbb Z$. This is what Manolescu disproved (there is no such 3-manifold); the Galewski-Stern paper clarifies what dimensions their theorem is proved for after you start dropping assumptions about compactness and the boundary.
But importantly for your question is Galewski and Stern's sequel paper. Their theorem 2.1 (plus Manolescu's result) implies that there are non-triangulable manifolds in every dimension $n \geq 5$. However, all orientable 5-dimensional manifolds are triangulable. In dimensions at least 6, though, you can use their construction to produce non-triangulable orientable manifolds.