I think the answer is "it depends"
(And I'm not familiar with national SIMs in those countries, so I can't help on specific -- but others here can)
Are you going to be using the phone mostly to receive calls from the US, make calls back to the US or to make or receive local calls while in a country? And what level of use in each country (light, medium, heavy)?
If people need to reach you, you'll need to know the number before you leave and an international SIM is probably best. An international SIM also means you'll only have one number for them to call, no matter where you are.
If you are mostly "phoning home" I've found the combination of United Mobile (Riiing) and Call Back World to work well, though I've no experience on how it works in Italy and Greece.
If you are going to making or receiving lots of national calls in a country, or you are going to be traveling there frequently then a national SIM might makes sense.
Another factor is how much you are going to be using the phone. If you get national SIMs for each country you are going to have to buy airtime for each. If you have to, for example, buy a minimum of ?15 in each country but only plan on making a few calls, an international SIM might be cheaper and have lest wasted airtime.
On more thing to consider is how often you'll be traveling outside the US. If you do a couple times a year, to many different countries then an international SIM makes sense. UM has to be used or topped up every 9 months (and an SMS counts as use) so its not hard to keep it alive for annual trips. And you can loan out your international SIM to friends who travel.
Sorry there aren't more specifics -- I'll leave that to the "experts" here on Italy, Greece and Austria.