Hello, I just got back from a non-cycling tour of Transylvania (3 days) and Budapest (1 week). Although I didn't ride them personally, the roads looked pretty good and there were locals riding old bikes (and walking!) along some of the highways in between towns in Transylvania.
I visited Brasov, Bran, Sinaia, Sibiu, and Sigisoara and I can say that I was totally comfortable the entire time as far as safety was concerned, including some pretty late night walking about Brasov and Sigisoara. I was followed by a few stray dogs (who seemed quite nice, although I didn't try to test this theory!) but that was the worst that happened. Haha, my friend and I did get some strange stares from locals though. I don't remember the beggars to have been any worse than most of the other places I visited (many, many places in E. and W. Europe) on my trip.
Stores were readily available in these towns and prices were pretty cheap. Also, Romanian is a Latin-based language so if you have any knowledge of other Latin-based languages (i.e. Spanish, Italian, French) than many words are easily understood. The Romanian Lei, for me at least, was pretty hassle free (and cool looking), except for the fact that I could get NO money out from Romanian ATMs (I didn't have this problem in any other country)!! Luckily my friend was there to bail me out.
Budapest is an awesome city and has its fair share of bikers. The Danube path looked really nice and although everybody I talked to said that Budapest was a dangerous city for bikes, it seemed no worse than any other big city. There are a good number of bike paths and enough bikers to where the drivers have some awareness of bikes. And I saw about 3 million bike messengers riding around the city in a crazy manner! Hope this was at least a bit helpful!