Having lived in Israel during several periods of my life, I know that the words "Old" in the "Old Testament" and "New" in the "New Testament" are offensive to many Jews. From what I understand, it's because "old" tends to have negative connotations (ex: outdated, irrelevant, decrepit, obsolete) and "new" has positive connotations (ex: new and improved, better, more relevant, contemporary, updated). I understand their logic and sentiment about all of that! In fact, I agree with them full-heartily. Let's be honest, if you were to say: "do you want this 'old' toy [any item, really] or this 'new' toy [any item, really]?" the majority of people will prefer the "new" item because we tend to associate "new" with "better quality," especially in comparison to anything "old" (unless you're an antique collector, historian, etc).
Thus, like Jews, I prefer to call the "Old Testament," the "Hebrew Bible" -- so it's free of all those negative connotations. This is not just because I want to be PC or demonstrate my respect/sensitives towards their perspectives (though both are true), but also, I think... even as a Christian, though we value both bodies of scriptures, the "old" in "Old Testament" carries that same negative baggage with it... "It's too old and obscure to understand... and irrelevant since Christ fulfilled the Law of Moses...." And I think this is a dangerous approach to this sacred literary corpus insomuch we develop preconceived notions that it will be too challenging to comprehend/apply and worse, insignificant (I don't think we do it consciously, it's just embedded in the word "old"). Consequently, we have an innate proclivity to value it less... And this is sad since there is much to be gleaned from the Hebrew Bible! From the Hebrew Bible, the divine covenants concerning salvation were taught as well as the Abrahamic covenant, etc. etc. Many of these principles and prophecies are timeless, and should not be burdened down by the word "old," which denigrates its worth and authority a bit.
Now, I am not naive, and I know the Christian communities will never adopt such a name change because it's just not practical nor a priority, mostly because most people don't care and a change in the name will just cause confusion, and cause more problems. To be clear, I'm not advocating/promoting that the Churches do so.... But, I must say, that I am pleased to see that "Hebrew Bible" is being used more and more in biblical scholarship and the academic spheres. Personally, I'm going to use it in lieu of "Old Testament," (even within a Christian context) and I will be happy to explain/elucidate my reasoning to whoever I'm speaking to.
However, after this long-winded spiel, I realize (and I'm sure you have by now, too) that I don't know what neutrally-charged, more appropriate, title to use when referencing the "New" Testament. I know within the walls of synagogues, the "New Testament" is not a likely topic that arises, but how do I reference it in an inter-faith academic dialogue if I don't want to pin on the "New" connotations? In acadamia/biblical scholarships, what name/title do they ascribe to the "New Testament." I'm assuming, they probably call it "New Testament" just so that everyone is on the same page and knows what they are talking about... but if we were to have a "Hebrew Bible" equivalent for the "New Testament," what would it likely be? "Greek Bible" doesn't work because the Septuagint (LXX) is the Hebrew Bible translated in Greek by Jews (hundreds of years before the existence of the "New Testament")... "Christian Bible" doesn't work because that seems to confine Christians to only the "New Testament," which only perpetuates the dilemma. Or is there already one, and I am just uninformed?
Basically, how do Jews refer to the "New" Testament? What is PC?