ETYMOLOGY:
Middle English rivere, from Anglo-Norman, from Vulgar Latin *rpria, from Latin, feminine of rprius, of a bank, from rpa, bank.

This puts me in mind of the interesting adjective form of "river" -- "riparian" (pron. ripARian). Someone whose land allows access to a river is said to have "riparian rights."

And back to Ants' origianl question, while I *feel like there is a "real" word for what you're thinking of, I'd like to propose the following coinage for those who'd like to be able to get back in that same old river: "homoriparian."