That depends on your home country.
Some countries are satisfied if you can show that you have simply cut all ties with your home country: no rented apartment, no real estate, no close family (wife/kids), no investments, (almost) no time spent in the country etc.
Countries from that category would be the UK or Germany, as far as I know.
Other countries also want to see proof that you have actually moved permanently: documents that you have rented a place in a new country, proof that you spent a lot of time there (utility bills, receipts, flight tickets, ...).
Otherwise they say you should still pay tax in your home country since you're just "traveling" and have not really left for good. Countries from that category would be Australia or Portugal, as far as I know.
Put yourself in the position of the tax inspector from your home country (Italy?):
Some guy tells you: "I have moved to Romania, I no longer want to pay tax here."
"Ok, please show us some documents that you have actually moved to Romania"
"Here's my rental contract"
"Ok, that's fine, but we need some proper proof that you actually live there, then I will delete you from our database of tax slaves"
"Errrr... Well, I actually only spent 5 days there since I moved"
"What do you mean? You just said you have moved there? Why didn't you spend time there"
"I spent 1 month in Italy, then 2 months Slovenia, then 2 months Spain, then I spent 5 days in Romania, then 2 weeks in Croatia, then another 2 weeks in Italy"
Now compare this case with an Italian engineer, living in Italy, with clients from all across Europe:
"I spent 1 month in Italy, then 2 months in Slovenia, then 2 months in Spain. A bit of it was for work, a bit of it was a vacation. Then I spent 5 days working in Romania and then 2 weeks' vacation in Croatia, before returning to Italy."
The latter person should clearly pay their taxes in Italy. But are you really that different from that person? That is what the tax authorities might wonder.
They might take the position that your center of life is still in Italy, especially if you keep returning to Italy every few weeks/months and spend only very little time in what you claim to be your new home.
It would probably be a different story if you could show that you spent 4-5 months in Romania (or at least outside the Schengen area), that you only spent 2-3 weeks in total in Italy and that you were traveling the rest of the time.
(They probably wouldn't be able to see if you were actually in Romania, just whether your were in the Schengen area or not.)
It would probably also be a different story if you said you have moved to Spain and they see that you have indeed spent more time in Spain than in Italy. But spending more time everywhere else than in what you claim is your new home? Maybe even more time in your home country? That sounds very strange...
But I have no idea how strict the rules of your home country are and how motivated the tax authorities are to look closer into a case like yours.
Maybe they would only ask to see the Romanian rental contract and be happy. Maybe they would only ask for a tax residency
certificate and be happy. Or maybe they would want to see a complete list of how many days you spent in which country, for the whole year. Or maybe they would pull Schengen immigration data to see if you're telling the truth.
If you want to get a proper answer, you should talk to a tax lawyer from your home country and ask how it's handled in practice.