This tip will work for any vent; I used this on our air intake vent (the most shockingly dirty vent in the house).
Below, my 2/3 cleaned vent is nice and white on the left side, and disturbingly grungy on the right side. I am convinced that you cannot appreciate the actual level of grime by looking at the picture. Believe me, it was obvious to anyone walking by.
So, how do you clean a vent without taking the whole grill off? My dear neighbor, who cleaned professionally at one point, taught me this trick.
First, gather cleaner of choice, a metal butter knife, and a cleaning rag.
Second, wrap the cleaning rag around the knife- if it is a thin rag, double the layer.
Third, spray the vent with cleaner (whatever you’re comfortable with, be it 409, or an eco-friendly non-toxic cleaner, or just water. Seriously, whatever floats your boat.) and insert knife between the metal slats. Clean side to side.
Fourth, readjust the rag as necessary so you have a clean area of the rag to use. Rinse and repeat.
Et voila, after less than five minutes: a vent that is spic and span!
This is a particularly good trick for vents that are very difficult to remove, or the paint would be damaged by removing.