We study the effects of the electronic coupling to bosonic modes on scanning tunneling microscopy (STM) into a d -wave superconductor. We propose to investigate these effects by means of a different technique: a Fourier transformed inelastic electron tunneling spectroscopy (FT-IETS). Specifically, in this technique, the Fourier spectrum of the energy derivative local density of states is addressed, which is proportional to the (d2 I d V2) (q,eV) characteristics measured in FT-IETS STM. We consider the role of the electron scattering due to a boson with the specific examples of the B1g phonon, the breathing mode phonon, and spin resonance at (π,π). It is found that the B1g mode with a highly anisotropic momentum-dependent coupling matrix element gives rise to well defined features in the Fourier spectrum, at the energy of mode plus gap, with a momentum transfer along the Cu-O bond direction of cuprates. This result is in striking contrast to the cases of the coupling to other modes and also to the case of no mode coupling. The origin of this difference is explored in detail. A comparison with the recent STM experiments is briefly discussed. © 2006 The American Physical Society.