Hi there.
There are 2 straps (and a carrying handle) made for the GX680s. The original strap for the GX680 I & II will not fit on a MkIII - this is the one that forms a cradle for the body and attaches to the tripod mount. The MkIIIs takes a conventional strap, though with bespoke lugs, attaching to lugs on the side of the body.
On a personal note, I wish the original strap/cradle had been modified for use with the MkIII as it is much easier (and feels more robust) for use when hand-holding the camera. That being said, the carrying handle is my favourite though it's not so good with the MkIII AE Finder or Magnifying Hood attached.