Let $M$ be a complete, connected, two-dimensional Riemannian manifold with nonpositive Gaussian curvature $K$. We say that $M$ satisfies the unrestricted complete controllability property for the Dubins' problem (UCC for short) if the following holds: Given any $(p_1,v_1)$ and $(p_2,v_2)$ in $TM$, there exists a curve $\gamma$ in $M$, with arbitrary small geodesic curvature, such that $\gamma$ connects $p_1$ to $p_2$ and, for $i=1,2$, $\dot\gamma$ is equal to $v_i$ at $p_i$. Property UCC is equivalent to the complete controllability of a family of control systems of Dubins' type, parameterized by the (arbitrary small) prescribed bound on the geodesic curvature. It is well-known that the Poincaré half-plane does not verify property UCC. In this paper, we provide a complete characterization of the surfaces $M$, with either uniformly negative or bounded curvature, that satisfy property UCC. More precisely, if $\sup_MK<0$ or $\inf_MK>-\infty$, we show that UCC holds if and only if $(i)$ $M$ is of the first kind or $(ii)$ the curvature satisfies a suitable integral decay condition at infinity.