We introduce a class of kinetic and anisotropic random motions $(x_t^σ,v_t^σ)_{t \geq 0}$ on the unit tangent bundle $T^1 \mathcal M$ of a general Riemannian manifold $(\mathcal M,g)$, where $σ$ is a positive parameter quantifying the amount of noise affecting the dynamics. As the latter goes to infinity, we then show that the time rescaled process $(x_{σ^2 t}^σ)_{t \geq 0}$ converges in law to an explicit anisotropic Brownian motion on $\mathcal M$. Our approach is essentially based on the strong mixing properties of the underlying velocity process and on rough paths techniques, allowing us to reduce the general case to its Euclidean analogue. Using these methods, we are able to recover a range of classical results.