Jenna,
Here in Ladakh we plant willows from cuttings all the time, but there are almost no weeping willows in the area. Weeping or not, willows will
root fine from cuttings. But I'm afraid you might be a week or two too late, now, in late April. Once the leaf buds start swelling out and showing green, in my
experience those cuttings don't root well. If you do it in earlier spring before any signs of green, stamp your cuttings into the ground well, and water frequently during the first season, there is a very high rate of success.
The climate is very dry here, but if willows get watered every 5 or 7 days they seem to do fine. Well, that's the willow varieties here. We pollard them every three years once they are established, and use the poles for tool handles, planting new willows, roofing sticks, and
firewood.