I am a U.S. citizen that travelled to Romania and overstayed my visit by about 12 days. I filled out paperwork to exit the country and was only given a warning because it was my first time overstaying. I am now back in the states and want to take a short vacation for a few days to Greece with my boyfriend. Will I run into any trouble? I know Romania isn’t a part of the Schengen zone, but will Greece see that I’ve overstayed my trip to Romania and not let me enter? What about if my flight lands in another country first to transfer to Greece?
What could happen is that the border guards see the Romanian stamps in your passport and decide that you are an unreliable traveller because you have previously shown your willingness to skirt the rules. They could then ask you to provide more details about your plans and/or ultimately rule that you were not able to establish you have a valid purpose for the stay (as a valid purpose obviously does not include overstaying and that's what you would be suspected of planning).
Transiting through a third country does not make any difference, legally (but if you transit elsewhere in the Schengen area, you might very well be denied entry then and there and sent back immediately, which would prevent you from even reaching Greece). Spending a longer period of time in your country of usual residence or having some documentation to show what you want to do in Greece and for how long might help. Going to visit your boyfriend (is he a Greek resident?) might theoretically play against you as you might tempted to overstay again to stay with him.
On the other hand, even a previous overstay in the Schengen area does not make it legally mandatory for border guards to deny entry (and technically isn't a motive to deny entry either, border guards have to evaluate your current intent and circumstances and base their decision on that, at least formally). It's also entirely possible that they don't notice or care all that much so it's impossible to predict that you will definitely be denied entry. However, Greek border guards are possibly somewhat more likely to notice Romanian stamps thasn stamps from any random country in the world because as part of its preparation to enter the Schengen area, Romania has to apply the same rules and uses stamps very similar to standard Schengen stamps.