If you're looking for a romantic get-away and willing to spend some money, you may want to travel to Cappadocia, Turkey. 

Cappadocia, Turkey is chock-full of cave dwellings, green valleys and one-of-a-kind hotels that are carved directly into cliffs.  Feast your eyes on  the Gamirasu Cave Hotel and the Rock Hotel.  Warning!  Viewing these hotels may cause you to book your flights today.

Note: I've never been to Turkey.