score:4
I was in Bucharest recently. It's a safe, pleasant city - and very tourist friendly.
I felt perfectly safe carrying my passport around in a buttoned down pocket. If you're nervous about losing it, I can suggest the following:
Upvote:3
I am a US citizen who spends two 90 day periods a year in Romania (3 months in US, 3 months in Romania, rinse, lather, repeat).
Never once have I been stopped, questioned or otherwise asked to see my passport by any law enforcement authorities other than the customs guards upon entry or exit to the country.
That being said, you should always carry some form of identification. A driver's license is more than adequate for identification purposes. Leave your passport in a safe place and only take it out when you are crossing the border or exchanging currency.