In the not so (yet) popular area of Boyalik. Cross the road, there is a beach. Do not cross it, jump in the various buses that take you to Cesme city centre and/or Alacati, Ilica... Very short trip and economical.
There is also a direct and bus from the bus station.
Rooms are a delight with sea views but in a populated area, so expect buildings in that frontal view too.
Really worth a try.