How does the Eurozone go about dealing with it's debt?

Like many countries in the euro, Greece has the potential to actually have credit risk, because it can no longer print the currency with which to pay off its debt. As ugly as debt levels are in the United States, Japan or other non-euro countries, downgrading their credit is silly, since they can pay what they owe by just printing the money, but those in the Eurozone are going to have bigger problems dealing with their debt.