set the UVs to their vertices's coordinates? Is there a way to unwrap the UV's so that they match the coordinates of their respective vertices in the XY, XZ and YZ plane?

I have a tiling texture and all my faces are aligned to either the XY, XZ or YZ plane. So, I basically want the texture tiles each unit in either direction.
To get your image to display once on each face, select the entire mesh and press U > Reset. Each face will now be mapped to the image bounds.

In the UV/Image Editor header turn on UV and Edit mode selection sync.

Now in the 3D view you can select any faces with the texture on the wrong way, and rotate them in the UV Editor.