May I recommend a cabin on the island of Kauai (in Hawaii)? If you look into this, try looking at Kahili Mountain Park, or Kokee. I can't imagine a cabin for a week there is too expensive, and it should give you the privacy you'd want on a honeymoon to do all those things honeymooners should be doing. Both Kahili and Kokee are kind of up in the mountains, but the beach is never more than 1/2-1 hour away, depending where your cabin ends up being.
This is what I've thought about doing on a honeymoon, just to get away from everything so me and my woman could focus on what we're supposed to be doing on a honeymoon. (Kauai is also where I'm from so thats why I can be so specific.)