A new, semi-spectral technique based on integral equations is introduced for the landslide tsunami problem. The technique does not use the shallow water approximation and resolves the dispersive surface wavefield generated by sliding material over a bathymetric profile. The wave scattering due to the bathymetric profile on which the slide occurs is calculated solving an integral equation. On the free surface, the linearized kinematic and dynamic boundary conditions are imposed. Method of images is adopted to solve for a source-sink distribution on the bathymetry to simulate the motion of the landslide and to satisfy the kinematic condition on the sea bottom. An asymptotic relation for the far field is also derived. The application to a landslide tsunami generation scenario in the Sea of Marmara reveals that a thickness H submarine mass failure on the southern rim of the Cinarcik Basin would create a wave peak of around 0.5 H on the similar to 1000 m deep Cinarcik Basin.