After years of driving in Romania, I would recommend :
Do not forget to buy a "rovinieta" (vignette, road tax stamp), even for one day : it is mandatory not only on highways, but also on many "national roads". The control is automatically done with camera systems, and you could have problems when leaving Romania to Bulgaria if you have not recorded your car. Take care also on Bulgaria : they do have a tax stamp system (like in most of eastern Europe countries). You can pay for these stamp at the border when entering in the country, or in gas stations.
Some have already said it : the heavy truck traffic can be a big issue, mostly if you drive through Transylvania which is a very mountainous area. Just let them overtake.
In countryside, you can meet, even on large road, some horse cars, many dogs (and other animals), bikes, and pedestrians (especially when crossing cities). Most of the time it will be OK because the road are clean, and the speed is limited. But if you drive at night, be careful. The exact same conditions apply for Bulgaria.
I have not seen any crime problem, nor border patrol or policemen asking for bribes. Just respect the speed limit, drive "normally" and everything will be OK.
People are very nice, and the countryside is amazingly beautiful. And because the country is still a bit "savage", some roads are magic like the Transfăgărășan which I recommend if you have time for it and if you drive around Sibiu.