Making them part of the landscape by adding a special terrain type doesn't sound possible, because they would have to be able to line up with each other, and if all the tiles have them on the left side (for instance) I do not think the game understands how to flip them around to make sure rivers actually connect going the other way.
Also, you would have to re-code the initial world generation scripts in order to ensure that a river is drawn continuously, and connects to a water tile at the end of it. Plus re-code the tiles to spread fresh water to neighboring tiles. And set them to provide trade routes.
The most likely possible solution would be to code them as a new road type, that cannot be pillaged, and somehow generates fresh water. But this means that you either reduce rivers to a single tile wide stretch of farmable land (the tile it is on), or you move to providing 3 tile wide stretches of farmable land (tile + neighbors). Also, you have to code the improvements AND resources to work around the river (cottages, Farms, Spices, Mithril...), otherwise you wind up cutting the river in half.
All said, you have to change a LOT if you move rivers off of the line between tiles. So while it might be possible to create something that looks like them.... for mechanics of the game, it is nearly impossible to generate anything that ACTS like them
But yes, there are tons of alternative rivers that would be awesome (like poisoned streams as a method to attack opponents using some really evil spells)