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 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.