I just got back from mine on June 1. By far, our best trip (other than our engagement trip) to the world ever! We stayed 10 days and split our stay between the Poly and POR. The Poly was gorgeous, more than I had expected it would be.
Disney gave us our "Magical Memories" pins and we wore them throughout the week. We got free desserts at Casey's, the Plaza restaurant pixie dusted our table, and we were sent to the front of the line and given free fast passes on numerous occasions. Over at Epcot, a CM handed us a phone and told us Mickey and Minnie wanted to congratulate us. It was a cute recording over the phone.
Everyone says congratulations to you and the pins also sparked up conversations with other guests while waiting in lines. It was fun to meet other couples who had honeymooned there and of course everyone offers you a little bit of marriage advice.
You can also get the bride/groom ears, but, they were a little uncomfortable in the heat. We only wore ours one day.
As far as dining, we had the dining plan so we ate at a different restaurant every night. I would recommend 'Ohanas and the California Grill as they are our two favorites and have eaten there on every trip.
For romance, I would suggest doing the Wishes fireworks cruise if you have the room to splurge a little bit. It was very romantic, just us and our driver on the lagoon. Or, something just as romantic and totally free, watch Wishes from the beach at the Poly. Very peaceful.
Sorry, that was long winded, but I could go on and on about our honeymoon!