Well, I don't want to stir things up any more, but I do have a rabbit who will be 7 this month, and doesn't eat hay.
Hazel has never liked grass hay, she refuses to eat it.
She gets lots of greens and vegetables, herbs and flowers, and a sprinkle of pellets as treats. I do dry home grown herbs for her, so she does have dry stuff, plus she loves the dried willow leaves and twigs, dried birch leaves and twigs, and dried grape vines and leaves.
She has had only one GI slowdown that needed vet treatment in all those 7 years, and that was last year when she turned 6. She recovered very quickly after being hydrated though. She has no tooth problems, never has.
I usually agree with Randy, but I must disagree that there is not enough roughage in greens. Things like kale, and fresh grass, are really good at wearing teeth down. Grass (and grass like plants) contain lots of silica, which is the component in hay that wears the teeth down.
Other foods rich in silica are cucumbers, lettuce, cabbage, asparagus, sunflower seeds, bamboo leaves and oat hay.
So if you are able to provide plenty of vegetables and fresh greens, as well as some dried vines and twigs, you can do it. You do need to have a plentifull supply year round though, and it really helps if you can grow your own grasses and herbs etc.