Results of numerical simulations of channeling of ultra-relativistic positrons are reported for straight and uniformly bent diamond crystals. The projectile trajectories in a crystal are computed using the newly developed module of the MBN Explorer package which simulates classical trajectories in a crystalline medium by integrating the relativistic equations of the motion with account for the interaction between the projectile and crystal atoms. A Monte Carlo approach is employed to sample the incoming positrons and to account for thermal vibrations of the crystal atoms. The channeling parameters and emission spectra of incident positrons with a projectile energy of 855 MeV along C(110) crystallographic planes are calculated for different curvature radii of the crystal. Two features of the emission spectrum associated with positron oscillations in a channel and synchrotron radiation are studied as a function of crystal curvature.