Are there any bike routes in Romania?

Google is your friend.    

A) Very few answer or respond to cycling questions - especially since Yahoo Answers is shutting down in a few days.  So, the chances of finding someone here with personal experience about Romania is infinitesimal - next to ZERO.  B) As with a lot of things in life, you stand a better chance of getting detailed info once you get there.  Ask LOCALLY!  C) A good place to start before you get there is so simple.  GOOGLE!  Google lists about 751,000 results.  That should get you started.  https://www.google.com/search?q=bike+routes+in+Rom...

Last...you NEVER stated whether you want to do this all on road, off road or a little of both.  Here's what one website describes as The 20 top rides in Romania. https://www.komoot.com/guide/72172/road-cycling-ro... 

Even in the International category Romania isn't listed. You might get better answers if you use the Edit icon to move your question to the Travel section. 

you might be able to find some online