I definitely wouldn't do it. Dish washer detergent has some powerful enzymes in it that will destroy a certain materials. If you ever put polished aluminum into a dishwasher, you'll find out in one wash just how corrosive that stuff can be. Not only that, but the dish washer tends to thrash stuff around a good bit, which is why you're not supposed to put knives in it. Not to mention the rapid heating which can ruin wood. You might be able to figure out a way to make it work. But I wouldn't waste my time trying.
If you have a bunch of filthy filters to clean at once, I'd just fill the sink up with warm, soapy water (I use Dawn, dishwashing liquid), and let them soak for an hour or so. Then wash them off gently with your hands underwater, rinse them in warm water, and dry them with some microfiber cloths (let them air dry a bit on their own first, so your microfiber cloths don't get soaked right away). I did that once with about 200 filters that I bought from a guy for next to nothing, and turned around and sold them on eBay. Most came out in like new condition (despite all of them being disgusting before the wash), and I'm pretty sure the few that had scratches, had those scratches from before (I didn't sell those).