Transmission grating-based x-ray interferometers have been developed into wide-field imaging devices that are sensitive to x-ray refraction and diffraction in the sample. While current grating designs rely on UV and x-ray lithography processes to produce periodic vertical structures, it becomes prohibitively difficult to make the grating periods below 1 - 2 microns due to the high aspect ratios of the structures. Since the phase-contrast sensitivity is inversely related to the grating period, we describe a new grating design for sub-micron to nanometer grating periods. In this design, multiple bi-layers of two alternating materials are deposited on a stair like substrate, and mostly on the floor surfaces of the steps only. The incident x-ray beam is parallel to the planes of the layers (side illumination). Thus, the multilayer structure on each step serves as a micro grating whose grating period is the thickness of a bi-layer. The array of micro gratings over the whole length of the stair can act as a single continuous grating, when certain continuity conditions between neighboring steps are met. Since the layer thickness can be as small as tens of nanometers, as has been demonstrated in multilayer x-ray zone plates, this design allows nanometer grating periods over large grating areas. Here we describe a prototype intensity grating of 440 nm period. We show x-ray projection images of the grating which were obtained by contact lithography.