Among the existent X-ray phase-contrast modalities, grating interferometry appears as a very promising technique for commercial applications, since it is compatible with conventional X-ray tubes and is robust from a mechanical point of view. However, since applications such as medical imaging and homeland security demand covering a considerable field of view, the fabrication of large-area gratings, which is known to be challenging and expensive, would be needed. A scanning setup is a good solution for this issue, because it uses cheaper line instead of large-area 2D detectors and, therefore, would require smaller gratings. In such a setup, the phase-retrieval using the conventional phase-stepping approach would be very slow, so having a faster method to record the signals becomes fundamental. To tackle this problem, we present a scanning-mode grating interferometer design, in which a grating is tilted to form Moiré fringes perpendicular to the grating lines. The sample is then translated along the fringes, so each line detector records a different phase step for each slice of the sample. This new approach was tested both in a simulated scenario and in an experimental setting, and its performance was quantitatively satisfactory compared to the traditional phase-stepping method and another existing scanning-mode technique.