Press a soapy mf sponge up against the grille, to force some water into the honeycombs and let the sponge's "fingers" get the majority of the dirt out. Then when it's all soaped up, rinse with a hose (spraying downward so you're not spraying your bay) and then dry it with a thick, plush mf towel in the same manner as washing: by pressing it into the grille. Works better if you do it regularly (obviously).
EDIT: Won't brake fluid eat at the plastic? If so and if the grille is real dirty, take it out, wash it and use a 10:1 mixture of water and iso alcohol to really spiffy it up. You might be able to use a slightly higher concentration, but don't go too high as I've seen it dull plastic (with straight up alcohol).