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.