The external Hartle-Thorne geometry, which describes the spacetime outside a slowly rotating compact star, is characterized by the gravitational mass M, angular momentum J, and quadrupole moment Q of the star and gives a convenient description, which, for the rotation frequencies of more than 95% of known pulsars, is sufficiently accurate for most purposes. We focus here on the motion of particles in these spacetimes, presenting a detailed systematic analysis of the frequency properties of radial and vertical epicyclic motion and of orbital motion. Our investigation is motivated by X-ray observations of binary systems containing a rotating neutron star that is accreting matter from its binary companion. In these systems, twin high-frequency quasi-periodic oscillations (QPOs) are sometimes observed with a frequency ratio approaching 3:2 or 5:4, and these may be explained by models involving the orbital and epicyclic frequencies of quasi-circular geodesic motion. In our analysis, we use realistic equations of state for the stellar matter and proceed in a self-consistent way, following the Hartle-Thorne approach in calculating both the corresponding values of Q, M, and J for the stellar model and the properties of the surrounding spacetime. Our results are then applied to a range of geodetical models for QPOs. A key feature of our study is that it implements the recently discovered universal relations among neutron-star parameters so that the results can be directly used for models with different masses M, radii R, and rotational frequencies f_(rot).