Most of the mode conversion theories considered so far assume only a plane-layered medium, i.e. a medium where the parameters depend on one spatial coordinate. We generalize the mode-conversion method of Cairns and Lashmore-Davies to plasmas with two-dimensional inhomogeneities. In the method presented here, the frequencies $\omega_1$ and $\omega_2$ of the uncoupled modes belonging to two different dispersion equations are considered as functions of the space variable $\mathbf r$ and the wave vector $\mathbf k$ and are coupled together via a small quantity $\eta$. We calculate the energy transmission and conversion coefficients analytically by solving two coupled wave amplitude equations in the electron cyclotron range of frequencies. The results are applicable to electron Bernstein wave heating of plasmas with two-dimensional inhomogeneity, e.g. spherical tokamaks.