Here's the lowdown on the places I've discovered so far:
The Pont des Arts, the pedestrian bridge over the Seine opposite the Académie Française is popular but relaxed. You can sit on the wooden boards watching the sunset over the river and wave at the people on the cruise boats down below. (We had a very international picnic here, followed dancing the tarantella at Place d'Italie and eating proper Italian ice cream. The Italian festival has finished now but I definitely recommend it for next year!)
The Ile de la Cité: the western quai of the island seems to be popular and looks very beautiful but one of my friends claims she saw a rat there once, so the Pont des Arts is probably safer!
The Bois de Vincennes turned out to be a less good place for a hike than my walking guidebook would have had me believe, but the Allée Royale has a stunning view of the castle and would be great for a big group picnic with lots of games.
The Terrasse de Saint-Germain-en-Laye also has amazing views, this time of the river Seine and the skyscrapers at La Défense. For variety, hiking, cycling and shade, there is also the enormous forest just behind.
The Champ de Mars is also supposed to be a cool place of an evening but I haven't been yet so I'll let you know!