I took cuttings of my brother's weeping willow tree in the late summer last year and they are starting to show some signs of life. I'm excited to see what will come of these. The Weeping Willow has always been one of my favorite trees.
Weeping willows are sooo easy to propagate by cuttings. Even branches as thick as 5 or 6 inches will take root after a couple of weeks in water. When the roots have formed, the water becomes a bit sticky: it's full of auxins, so you can use it for other cuttings or air-layering with any other species. I'm pretty sure it's been discussed here before. And search Harry Harrington's website, he wrote an article about that.
The pb with weeping willows is that they don't heal well, and also that when you prune them when they're in dormancy, die-back of sections are very likely. that's why I think pruning them when in full leaves is better.