Post-US currencies?

I was up late talking to my modmates last night about the existence of the U.S. government, and one of the things that came up was the continued use of U.S. currency. If the United States dissolved, what currency would the subsequent independent states use? 

I didn't have a good answer for this. My first thought was to wonder whether the federal government could be retained with no function other than managing a currency, to avoid financial catastrophe while the new states get their feet under them, financially. But this isn't an unambiguously good idea -- one of my modmates cited the issues with the Euro, and complications implicit in a shared currency between states that don't have a shared economic policy.

My next thought, which I didn't say out loud because it felt even less-well-developed than that first one, was some kind of cryptocurrency, like bitcoin. (Not actually bitcoin, though. Bitcoin is terrible.) I don't actually want to make this argument because I wouldn't be willing to claim with confidence that a currency that doesn't have a government could ever stabilize enough to be functional.

I'm not even sure how to start thinking about the process of 50-ish new currencies emerging at the same time out of a previously interdependent economic system. Hopefully I'll have more detailed thoughts on it soon.