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 did the same, perhaps you can imagine my horror when a buddy of mine said he hadn't had bananas before and every day after school his mom would make him and his brother a pizza.