Try switching your image texture node from flat to box mapping and use the Geometry -> Position as vector. You will need to add a mapping node to control the scale to get it right for your scene.
Thanks for your response. I tried your hint but there was still something wrong. In “Material” display method I could still see only faces in one axis mapped correctly, the others were still flat. But then I tried to switch to rendered mode… and Voila! The desired result was there! This is really annoying about Blender… But thanks a lot for solving my issue!