I think there are probably 3 general groups: those with essentially no savings, those with significant savings, and those whose savings exceed FDIC insurance limits. I also think the right thing to do varies significantly depending on which of these groups you fall into.
Those with more $$$ than the FDIC limits will probably want to diversify the cash they have on hand, or move it someplace really safe (Switzerland?).
Those with a lot of cash but under the limits will need to decide what they want to do with their money. If it is long term savings (5+ years) they can probably just stay pat. If the bank fails they should get it back eventually. If the savings is for a large shorter-term purchase (house/car/boat), then I'm not really sure what you can do.
Either way, these first two groups can probably make due in day-to-day expenditures using their credit limits, because they probably live at/below their means anyways.
Those with almost no savings might not be remiss to bury a couple hundred in the ground somewhere, since if they have their last check in the bank, they probably can't wait another week to get access to cash to pay their bills.