A well-known result of Hedlund and Morse states that if a Riemannian metric on a closed surface of genus $g > 1$ has no conjugate points, then it carries transitive geodesics (i.e., geodesics whose velocity vectors are dense in the unit tangent bundle). 


*Are there Riemannian metrics on closed surfaces of genus $g > 1$ that do not carry a transitive geodesic and if so what is the weakest condition known under which the existence of transitive geodesics has been proved?*


**Addendum.** As Misha remarks in his answer, it is easy to construct to metrics on any closed surface that do not carry a transitive geodesic. However, the metrics I'm interested in have the additional property that *the lifted metric on the universal cover has no trapped geodesics.* In other words, no geodesic stays forever in a compact subset of the open unit disc.



Transitive geodesics on closed surfaces of genus greater than one
This is both: answer on the new version of the question and comment on comment of Andrey Gogolev, who asked whether one can make the question more complicated assuming 
 additionally that the set of non-periodic geodesics is dense. 


Hier is an example that answers both: it is not much different from the answer of Andrey from his comment (and also from katz's answer). 


Take a torus with periodic coordinates $x,y$ and 
 a liouville metric $(X(x) -Y(y)) (dx^2 + dy^2)$ on it (whose geodesic flow is integrable). Make two small holes on the torus near the points where $X$ has minimum and $Y$ has maximum and connect by a neck. For generic $X$ and $Y$ and generic neck the set of nonperiodic geodesic is dence but still it has a 'regular' region in the tangent space where no geodesic passing through at least one point of neck can come. The closure of a typical geodesic from the regular region is a torus in the tangent space which is projected to an annulus on the torus and after passing to the universal cover this annulus becomes an infinite unboundend band. 


One more remark is that 
by KAM theory, this examples survives if you slightly perturbe the metric. 


