My vest is a lot more simple than the majority of the users here (Pilot), but I just took the padded parts and the velcro parts off, then
1) put them in the tub filled with cold water, pressed them by hand for a few minutes, let the water down.
2) used the most delicate detergent I found at home (made for breathable products, brand: Storm), used about 2 cups in the water, pressed the pads all over for about 10 minutes (wanted to do a thorough job as my gear was a rental kit), let them in water for about an hour more, let the water down. (the sewn 'Steadicam' logo was quite dirty, I have put a lot of delicate concentrated cleaner spray (also from an outdoor shop) on it and gently rubbed in with my fingers, the logo is beautiful white now)
3) rinsed them thoroughly, dried the parts outdoors by wind&sun for 36 hours.
Worked like a charm.