EDIT: I had to fragment the URL because I’m not allowed to post a single link here. Don’t know why… Sorry for that.
EDIT2: Some ideas. Is something like this possible?
a) After grouping the chess pieces, can I assign one single material to the whole group?
b) Can I somehow join the objects to one single object (and then apply my material)?
I figured out a way how to render a correct heightmap with multiple objects.
You have to remove any linked material on every object. Then join the objects by selecting and hitting CTRL+J. Now you can assign one material to the joined objects like in the tutorial mentioned before.