Good discussion Oceander, EC and Alice. Was on jury duty yesterday and missed the real discussion on the thread. I do find the concept of the IOU note interesting, but there would be some issues, namely what backs it up other than trust. California tried that and banks stopped taking them, and that was just a budget crisis, not a currency crisis.
Not having access to currency is worse than just printing tons of currency, so that's what the government would have to do, even at the risk of hyperinflation. We can once again talk about "not being worth a continental". And obviously the government would have to keep the banks afloat to distribute the monopoly money. The only real faith people have in our government is based on a viable system of currency IMHO. The rest of what Uncle Sam does is still supported by money.
Interestingly, the dollar is continuing to strengthen against the Euro. Is that less of a good sign for us or more of a bad sign for the EU? 
There are issues with private notes, including private bank notes as well as IOUs. Most of those problems don't show up in small, closed economies, however. That is how the concept of negotiable instruments developed. For the most part regular people no longer use many negotiable instruments; the exception being personal checks, but even then personal checks are usually only negotiated once - when they're deposited or cashed at a bank. Technically speaking, a personal check can be used to pay another third party - provided that third party will accept it - if the holder endorses the check and hands it to the third party; in fact, they can technically be negotiated multiple times, with each holder endorsing it at the time it's negotiated.
Warehouse receipts are another form of scrip currency that used to be used but aren't much anymore.
Basically, the system of fiat paper currency we have now is of very recent vintage and has only a few historical precursors. Most currency systems beforehand were usually systems of privately issued notes - banks typically issued them because banks held a fungible asset that could be readily traded in place of the bank's notes - gold and silver - but the colonies, and then the states - prior to the Constitution - also issued various scrip currencies; in the agricultural states scrip would be issued to farmers and collateralized with the farmer's crops in the field. In fact, one of the goals of the Constitutional Convention was doing away with state-issued scrip, in part because many of the states that issued scrip also tended to try and force creditors, especially out of state creditors, to accept that scrip in full satisfaction of the debts owed to those creditors. Since out of state creditors lent in hard currency, they expected to be repaid in hard currency, and state-issued scrip was basically useless to them. Hence the prohibition in the Constitution on the states making anything but gold or silver money, and the power granted to Congress to control the value of money.
