What methods are being used for cystocele repair since mesh was found to have problems?

Most repairs can use. Your own tissue and stitches. Mesh is still appropriate in some cases. When used by an experienced gyn surgeon or urogynecologist it has reasonable risks.
Traditional repairs. Vaginally placed mesh is the mesh you refer to. The area of dissection by surgeons who do a lot of this type of surgery has changed since the development of mesh. We are now doing the traditional repairs with "full thickness" dissection that was learned with the development of meshes. The jury is still out on the long term objective data regarding cure rates with all of the cystocele repairs.