Various schemes for electron cyclotron resonance heating of tokamak plasmas with the ratio of electron plasma frequency to electron cyclotron frequency, "»pe/^ce* larger than 1 on axis, are investigated. In particular, a mode conversion scheme is investigated using ordinary waves at the fundamental of the electron cyclotron frequency. These are injected obliquely from the outside of the tokamak near an optimal angle to the magnetic field lines. This method involves two mode conversions. The ordinary waves are converted into extraordinary waves near the plasma cut-off layer. The extraordinary waves are subsequently converted into electrostatic electron Bernstein waves at the upper hybrid resonance layer, and the Bernstein waves are completely absorbed close to the plasma centre. Results are presented from ray-tracinq calculations in full three-dimensional geometry using the dispersion function for a hot non-relativistic plasma. Radial profiles for the power deposition and the non-inductive wave-driven current due to the Bernstein waves are calculated for realistic antenna radiation patterns with parameters corresponding to the Danish tokamak DANTE and to Princeton's PLT.