Are European Banks Safe For Depositors ?

Since the "financial crisis"* started, I never felt that insured bank depositors were ever at any risk. Now there are reports that Greek citizens are withdrawing huge amounts of Euros from their local banks.
Their fear ? That they wake up one day and hold rapidly devaluing Drachmas instead of Euros.
Similar thinking seems to spread to other parts of the distressed Euro zone:
According to one of todays headlines, Irish residents trimmed their deposits in their banks by 9.1% year over year.

*Financial crisis is a misnomer: It is a bankruptcy that is not allowed to happen. In trying to avoid outright default, governments in the USA and Europe are, unfortunately, making the problem bigger and the eventual default much more painful.
How will the default manifest ? When the governments involved will issue intrinsically worthless paper to pay for their debts.