I can't speak to those cushion covers in particular, but if you do decide to wash them in a machine, I would recommend using one of the front-loading type machines that do not have the agitator in the center.
Years ago I washed a sleeping bag in one of the agitator-style machines and it was quite hard on it (rips). Many subsequent washings in the front-style with no problems, so I know they're gentler on that sort of thing.
PS: I would probably "hand" wash them myself, at least for a first round (only going to machine if I had to). I say "hand" because I would probably put them in a bathtub or large plastic garbage can with some soapy water and do "The Stomp."