Hello, and welcome to our forum community.
The question you ask has come up before - you might want to browse the
Europe section of the forum for more information.
Based on my experience touring Eastern Europe (for many summers, through all the various countries there), you really don't have much to worry about. Generally speaking, "tourist security" correlates directly with the level of economic security that the residents have. If you are in a country - or a city, or areas of a city - where the residents don't have to worry about their own economic security, they you won't have to worry about your personal security. The converse is equally true.
I have enjoyed my visits to Romania. Some parts of larger cities are dodgy, in the same way that some parts of larger cities in the USA are dodgy. Romania has an ethnic underclass (the Roma, or gypsies) who don't enjoy the same success and living conditions in cities that the majority of the population do. If someone has an empty belly, they are going to do what they have to do to get their next meal.
So, be aware of the 'lay of the land' in big cities, the same way you would be aware of the lay of the land in big cities in the Netherlands (I'm sure there are probably parts of Amsterdam or Rotterdam that are best avoided by tourists, especially at night).
In the rural parts of Romania, which I think are the best parts of the country, you don't have to be too concerned. Your biggest risk comes from traffic accidents, not from fear of assault or robbery.
Michael
Here are links to prior discussions of the same question in the Europe sub-forum:
Bike security when staying at hotels
Romania
Need info about riding in Romania
and finally, here is a link to a write-up I made about that region a few years ago:
Notes from a September 2011 ride through Romania, Bulgaria, Turkey, and Greece