Having one's own currency doesn't simply solve the problem. Plenty of Latin American countries know this first hand.

Basically, having your own currency allows you to print however much money you want to print. By issuing more currency, you as the state get to take resources from people without having to tax them. As the number of dollars increases, the value of each dollar decreases.* If you do this too much, you get hyperinflation. Because you are printing so much money, the value of that money drops at a ridiculous rate. I actually had an econ professor in college from Peru who said that during the worst days, a shopkeeper would close shop to get sales into the bank because if they waited until the end of the day, they would have lost so much value. Car companies would want to not sell cars, because the cars held their value more than currency did - so much for losing money when you drive it off the lot.

It is really hard to break out of a cycle of high inflation. Once the trust in the system is lost, it can be really hard to get it back. Many countries, in order to rid themselves of this hyperinflation, start using the currencies of other countries that manage their currency properly. For example, Panama, Equador, and El Salvador all use the US Dollar as their currency. Zimbabwe has started using a mixture of foreign currencies for trade. That eliminates the chance for them to print more money when they need and they have to rely on any loans they might get - which would be minimal given their credit ratings.

A devaluation of your currency and the inflation that come with it like that isn't good. It does allow you to get out of certain debts more easily, but it also means that investors don't want to invest or lend to you. If you have a history of using a trick to get out of your obligations, lenders don't want to lend to you. For most intents and purposes, such a huge devaluation is a sovereign default - which is what people are worried about with Greece. If you were able to pay back a fraction of your mortgage (say, 10%) through some devaluation, it wouldn't be a default, but it would mean that the bank was getting back a fraction of the money they had given you - even if the nominal value was the same. So, from the bank's perspective, it means they didn't get paid back even if they nominally did. (And if there were a way for individuals to do this, banks would treat it as harshly as a bankruptcy when making future credit decisions).

So, this wouldn't even be good for Greece. It might have meant that Greece would have seen inflation of 20%, 50%, 1000% per year for the past decade and their economy would have faltered along the way rather than them being able to hide the debt through funky accounting, but it wouldn't be a good option. Likewise, being able to print extra money isn't a boon for states either. Government revenue should come from taxes. A little can come from money printing in a way that is ok,* but taxes need to be the real source of government revenue. If you spend much more than you take in (a certain amount of debt for investment can be quite wise), you're going to get into trouble whether you can print your own money or not.

* In a manner of speaking. If the number of dollars increases at the rate of economic increase, the value of each dollar stays exactly the same (and helps protect against deflation). It's a smaller fraction of the pie (economy), but the pie is now larger.