This thread is a two-parter: general and specific discussion.
First the specific: when I was in Italy two years ago with my parents, our host’s parents wondered if I might come back sometime. Seeing as I failed to land a job this summer, I’m dying to go back to Italy. My (distant) relatives are fantastic people, and also, hey, it’s Italy. I haven’t communicated with them since my trip two years ago. Calling up and saying ‘hey how’s it going can I live in your house for two weeks during the summer’ seems awkward. How do you suggest I go about inviting myself over? 
Second, general discussion: staying in friends’/relatives’/associates’ homes is clearly the cheapest and best way to vacation internationally. Tell us how you’ve convinced otherwise obscure associates to have you stay with them. Or, tell us how you’ve gone about accepting an “open invitation.”
I dunno, it’s Italy. Italians often seem like Indians, warm & welcoming. I’ve called up my Indian relatives and said “I’m coming to India” and they all fight over who gets to keep me! And I can’t see Italians being less welcoming, on the contrary, they should be even more.
Can’t you drop the idea on them (you’ll be visiting Italy) and see how they take it?
Start with calling them, telling them that you’re planning a trip to Italy over the summer and wanted their advice about where you should go, what you should see and do, and when is a good time to be there weather-wise. That’s their cue to reiterate the offer to have you stay with them. You say, that would be great – I haven’t made any specific plans yet, and would love to stay with you. Then you sound them out about making their place your “base of operations.” That will let you know if they’re willing to put you up for 2 weeks, or if a weekend is more their style.
Honestly, I’ve found that letting people know that you’re going to be in their neck of the woods prompts most people to issue an invitation if they are willing to have you stay. It’s really just that simple. It’s how I’ve done it, and how others have done it to me (including my brother, notable for calling at 5 o’clock and asking, “what are you doing tonight?” then moving in for a week).