My husband and I really enjoyed our stay at Koza Cave Hotel. We had a junior suite, which was spacious but cozy, and the room was clean and decorated nicely. The guy who runs the concierge is incredibly nice and helpful. He also speaks English quite well, which made coordinating easy. There are also two beautiful long-haired cats who live on the property, Mimi and Kiki, who we enjoyed interacting with :). You really can't beat the views from the top deck, either, which provides a panoramic of Göreme.
Two things I would note:
-In our room at least (room 1), the bathroom was closed off from the rest of the suite using a curtain only, which means a lack of privacy when using the restroom. This is due to regulations surrounding modifying these historic caves.
2. Walking back up to the hotel from the little downtown area can get a bit tiring as in involves some steep inclines. It's good exercise, but I might suggest taking a cab ride up depending on your age/abilities.
All in all, we had a wonderful stay, and would stay again in a heartbeat!