We keep fresh fruits and vegetables sitting out on the counter at all times. The kids know the can munch on them without asking.
Grape tomatoes, oranges, bananas, apples, even the evil broccoli.
By giving them the options and variety of veggies to snack on whenever they want, it encourages healthy eating and good decision making. It also prevents them from becoming picky eaters that refuse to eat vegetables.
Edit: This also promotes independence and self reliance, so I'm not making them snacks and fucking sandwiches all the time. You want a snack? Make it yourself, bitch, I ain't your slave.
My parents always did the same. It helped we had a garden and several fruit trees in the yard, but on top of that there would always be some sort of melon cut up in the fridge (if in season), an abundance of cabbage, and my dad would make a little loose veggie bar between lunch and dinner most nights for snacks. It certainly rubbed off on me