Yep, both my dogs also occasionally graze. I've been told that the grass actually has a similiar taste to lamb, which some dogs find familiar to the foods that we already feed them. I wouldn't know, as my dogs certainly don't clarify this for me.
The only thing I want to precaution is to ensure that your grass-loving pup doesn't eat on lawns that you don't personally own. The city, and often many people without kids or pets, will fertilize and use chemicals which CAN make your puppy sick.