We have been using cloth diapers for nearly 2 years now. I tried to use cloth wipes a few different times but without much success, probably because we were using just baby washcloths that were not super soft. We started using cloth wipes exclusively in the last few months, and I do not see us going back to disposables.
The two main reasons for the switch were 1) with two in diapers we were going through a ton of wipes and 2) the disposable wipes seem to be giving our daughter a rash.
The cloth wipes are WAY better for newborn poop, which is really wet and runny. The wipes absorb more, instead of just pushing it around! A nurse at our pediatrician's office suggested using water and a cloth instead of the alcohol based disposable wipes, and this really helped to keep the rash away too! We keep a spray bottle on the change table and just spray a wipe before using it.
Our cloth wipes are nothing fancy; just some fleece cut into small pieces. We cut ours into two different sizes. The wipes are the pile on the right. And the pile on the left is our liners. We put the liners in our daughter's diapers for two reasons. If she has a poopy diaper, the fleece liner makes it way easier to get the poop off. Also, if we need to use some diaper cream, the fleece acts as a barrier. We just wash the liners with cream in a load separate from our diapers.
You can purchase cloth wipes and liners made out of fleece, but our simple fleece works so well that I don't see any need to buy something fancier.
The liners just lay in the diapers!
If you cloth diaper, cloth wipes are the easy way to go! And if you use disposable diapers, but have issues with rashes, cloth wipes (and diapers!) are a great alternative.