I have a beautiful collection of Japanese maneki neko cats on my china cabinet. I've collected these little cats from all over, and most of the time I enjoy looking at them. The other day, though, an errant ray of sunshine caught the shelf and I was shocked by how tacky they suddenly looked. Not because they are tacky, but because of the deep layer of winter dust that had accumulated on top each one.
Regardless of the type of ornaments you have, dust is a constant enemy. With my nekos I am lucky since each is made of ceramic. I can simply wash them in warm water without any fear of damage. Not all ornaments can survive this treatment.
Dust ornaments that contain fabric, feathers or silk flowers with a plastic bag and a small handful of rice. Place the ornament in the bag, hold it closed, and shake it around. The rise knocks the dust right off. Use your best judgment when determining if your ornaments can survive this treatment.
If you are afraid to get the ornament wet, especially if the paint can wash off, use a soft bristled small tipped paint brush to dust out the details. A lint-free buffing cloth can help dislodge deeper layer of dust without water or soap.
Finally, if you don't display the ornaments all the time keep them under wraps so they don't get dusty in storage. Place them in a box or carefully wrap them in paper, cloth or plastic. Dusting those on display frequently will also prevent a heavier, harder to remove buildup.