Nonlinear interferometers allow spectroscopy in the mid-infrared range by detecting correlated visible light, for which non-cooled detectors with higher specific detectivity and lower dark count rates are available. We present a new approach for the registration of spectral information, which combines a nonlinear interferometer using non-degenerate spontaneous parametric down-conversion (SPDC) with a Fourier-transform spectroscopy concept. In order to increase the spectral coverage, we use broadband non-collinear SPDC in periodically poled LiNbO3. Without the need for spectrally selective detection, continuous spectra with a spectral bandwidth of more than 100 cm-1 are achieved. We demonstrate transmission spectra of a polypropylene sample measured with 6 cm-1 resolution in the spectral range between 3.2 µm to 3.9 µm.