We adapt the Newman-Penrose formalism in general relativity to the setting of three-dimensional Riemannian geometry, and prove the following results. Given a Riemannian 3-manifold without boundary and a smooth unit vector field \(X\) with geodesic flow, if an integral curve of \(X\) is hypersurface-orthogonal at a point, then it is so at every point along that curve. Furthermore, if \(X\) is complete, hypersurface-orthogonal, and satisfies \(Ric(X,X) \geq 0\), then the divergence of \(X\) must be nonnegative. As an app- lication, we show that when \(Ric > 0\), a geodesic and divergence-free unit vector field cannot be hypersurface-orthogonal; the case \(Ric < 0\) yields known results pertaining to unit length Killing vector fields. Along the way, we connect this formalism to some recent results from contact geometry, and mention how the Newman-Penrose formalism may be used to help classify scalar-flat Riemannian 3-manifolds.