Look at these photos. You can still do this kind of thing, probably in more places than you’d think. One of them is the Klarälven River in Sweden. The outfitters from a company called Vildmark will take you out to the woods, show you how to lash up your raft, and then give you the heave ho.
Depending on the length of your trip, they’ll pick you up 50 or 100 kms down river. An eight-day, 50km trip starts at 3180 SKr (about $300) per person. Sounds like magic.