I've just switched to using bamboo dish cloths, Lakeland do one which is a thick, strong "stockinette" type and Amazon have a bamboo waffle fabric. Bamboo is supposed to be anti-bacterial so I wouldn't worry about putting them in with the normal wash.
The cloths are fairly expensive but the reviews say that they wash well.
I'll just have to see which last better without looking tatty.
I might put floor cloths through with a dark wash, but usually I pour boiling water over them.
I don't use cloths to wipe any chopping boards that have had raw meat on them, always use neat washing up liquid and a stiff brush. So my dish cloths are unlikely to be dangerously grubby.