I'll probably get flamed for being gauche, but nothing screams summer barbecue for me more than a big iceberg based, classic chopped salad. Something about the crunch and coolness just offsets the hot weather perfectly.
My very favorite is the Texas salad -- just chopped iceberg, chopped tomatoes, ranch dressing, and nothing else.
Doesn't satisfy the "green vegetable" requirement, technically, but it's the best salad ever, esp. in summertime.
Just sliced cucumbers drizzled with vinegar, salt and pepper is another one. Or for that matter, just sliced cucumbers and sliced tomatoes, period.