Is that not a thumbscrew on the left for removing the grille? Or yes it is, but the wood frame blocks removal?

If it can't come out, and vacuuming doesn't work, you could blast it with compressed air. It for sure will make a mess, and you'll have to clean your filter after, but it should work.

Unless that is greasy dust that sticks onto the surface then I think you're looking at using a damp cloth (or toothbrush like Mysterious-Ad7019 mentioned) and going 1 by 1 to clean them out.