Scattering of a transverse electric (TE) wavefield by an inhomogeneous lossy dielectric cylindrical obstacle embedded in a lossy dielectric half-space is investigated. The cylinder is of arbitrary cross-section, and its axis lies parallel to the interface. The formulation involves a domain integral equation of the field obtained by applying the Green's theorem to the appropriate Helmholtz wave equations. The convolution-correlation structure of the formulation is such that the discretized counterpart of the integral equation (provided by the application of a Method of Moments where the obstacle is divided into triangular patches and where the contrast and the H-field are expanded using two different basis functions) can be efficiently handled using a FFT-based conjugate gradient solver. The validity of the approach is illustrated by comparison with results found in the literature.