  We develop a numerical scheme for obtaining the $r$-mode oscillations of
rapidly rotating relativistic stars. In the present scheme, we neglect all
metric perturbations and only take account of the dynamics of the fluid in the
background spacetime of the unperturbed star (the relativistic Cowling
approximation). We also assume the star is barotropic, i.e. neutrally stable
against convection under the assumption of adiabatic oscillations. Our
numerical scheme is based on the Yoshida-Eriguchi formulation for the analysis
of the general relativistic f-mode oscillations in the Cowling approximation
and a general relativistic generalization of the
Karino-Yoshida-Yoshida-Eriguchi's numerical scheme for obtaining oscillations
of rapidly rotating Newtonian stars. By this new numerical scheme, frequencies
of the $r$-mode oscillations are obtained as functions of the ratio of the
rotational energy to the absolute value of the gravitational energy $T/|W|$
along sequences of polytropic equilibrium stars whose ratio of the pressure to
the total energy density at the center of the star and the polytropic index are
kept constant. It is found that the dimensionless oscillation frequency
$\sigma/\Omega$ is a linearly decreasing function of $T/|W|$, where $\sigma$
and $\Omega$ are the oscillation frequency and the angular velocity of the star
measured in an inertial frame at spatial infinity. We also find that
oscillation frequencies of the $r$-modes are highly dependent on the
relativistic factor $M/R$ of the star as already found in previous studies in
which the slow rotation approximation has been used. Here $M$ and $R$ denote
the mass and radius of the star, respectively.
