That's how I wash it after EVERY use, the included brush is worth it weight in gold for cleaning the screen. Otherwise the micro pieces of pulp will turn into cement and you'll end up with less and less juice. I use my juicer once or twice a week. About once a month, when I have a full load of dishes, I run all of the pieces of the juicer through my dishwasher to help sterilize it and get into some of the places that are harder to reach with either the brush or a sponge. Go forth and juice!
I hand wash mine. You can take it all apart (blade, juicer shoot ect....) It's actually better to hand wash because of all the left over pieces of fruit that is shredded. It's fantastic I use it a lot, and the blade replacements are cheap. You can purchase it at Kmart and I think Wal-mart-if you don't want to wait to have it shipped.