We analyse the geodesics' dynamics in cylindrically symmetric vacuum spacetimes with Λ > 0 and compare it to the Λ = 0 and Λ < 0 cases. When Λ > 0 there are two singularities in the metric which brings new qualitative features to the dynamics.We find that Λ = 0 planar timelike confined geodesics are unstable against the introduction of a sufficiently large Λ, in the sense that the bounded orbits become unbounded. In turn, any non-planar radially bounded geodesics are stable for any positive Λ.We construct global non-singular static vacuum spacetimes in cylindrical symmetry with Λ > 0 by matching the Linet-Tian metric with two appropriate sources.