This is a sortable table showing all the games which have been given away as part of Games with Gold. Useful to check if you cannot remember which ones you bought without downloading!
Note that you can buy an xbox One game if you only have a 360, in case you might buy in the future. Just go you xbox.com and buy the free game on there. There difference with Xbox One games with gold, is that you must have an active Gold subscription to play the game. With the 360, once you have the game, you can play it regardless.