Update: Japhy has also written on this topic, covering some additional ground.
In some circles there is a movement to reconsider the use of the term "Old Testament." This change is advocated by some within Judaism because of its implications that the New Testament supersedes the Old Testament. The replacement terms suggested, "Hebrew Bible" and "Christian Bible," are in theory supposed to be confessionally neutral. But each set of terms carries with it a set of confessional suppositions.
The term "Hebrew Bible" implies that those books are for the Hebrews and is a closed set of books for the Hebrews; it rejects the idea of a New Testament that is for the Hebrews as well. The term "Christian Bible" implies that those books belong to "another religion" (an idea the Jewish authors of those books rejected); it implies those books do not apply to Jews or Muslims or Buddhists. It bypasses the good news of God's covenant for the whole world, a new covenant which is not merely for all nations other than Israel, but a new covenant which includes Israel as the firstborn among the nations. "Hebrew Bible and Christian Bible" confesses religious pluralism and sidelines the idea that God might have revealed himself to the whole world though the Jewish Messiah; the terms marginalize any thought of a truth claim in favor of a comfortable "proprietary Scripture" formula. It establishes a sort of religious non-interference pact whose price tag is the universality of God's message and the brotherhood of all mankind under God's new covenant. It also insulates the Hebrew Bible from the idea that there might be fulfillment of its promises in the Messiah who died, the Messiah who suffered with his people, the Messiah who was both a light to the Gentiles and a glory to his own people, Israel. It pigeonholes Jesus as a merely partisan question rather than as the incarnate, living and breathing Torah of God. If I were to use the term "Hebrew Bible" for the one set of books, I would use the term "Worldwide Bible" for the other, since it reveals the Torah going forth from Jerusalem into all the world under the banner of the Messiah.
I have some sympathy with the distaste for calling a collection of Scriptures the "Old" Testament. If I were to avoid using "Old" Testament, I might say "First Testament" or "Early Testaments". True enough about the "old" part: nobody has performed the morning and evening sacrifices since the Romans demolished the Temple in 70 A.D.; there is definitely something old about it not merely in anciency but also in terms of non-survival to the modern world. The annual festivals have not been observed as prescribed by the Torah in nearly 2000 years; it has become impossible to keep the Torah, and no sacrifice for sins is left. And it would probably be in bad taste to confront a Jewish person with the fact that Jesus predicted the fall of the Temple as judgment because the leaders of the day rejected him. Although saying so is factually correct, it misses the most important point: not that Jesus was the downfall of the Temple, but that he was the fulfillment of the Temple promise: the place where God meets man, the place where God hears prayers, the place of forgiveness, the beginning of a covenant for all nations. If we tell Jews about Jesus as the one who was the end of the Old Covenant, they will meet him with resentment; they will accuse their brothers who follow the Messiah of "converting to another religion".
Better if we explain what the Torah did when the Word of God became flesh and tabernacled among us as Jesus the Messiah. It was Jews who took the Torah forth from Jerusalem under the banner of the Messiah. They took it to Egypt and the idols fell; Isis and Osiris and Horus and Ra became memories of the past. The Word of God did this, and Jews under the banner of the Messiah accomplished this thing. It was Jews under the banner of the Messiah who stopped the Assyrians and all the neighboring peoples from worshiping their false gods. The people who once led Israel astray were now cheerfully abandoning their false gods. The Word of God went to Greece and Rome under the banner of the Messiah, and Apollo and Zeus and Aphrodite and Hera became forgotten idols. Again it was Jews who did this, Jews under the banner of the Messiah. When the Jews list the accomplishments of their people, do they remember this?
Better if we explain that God's Messiah, like God's firstborn nation, shared in suffering. Better if we explain that here is real proof of Jesus' Jewishness when he shares in the fate of his people: everybody likes his ideas but nobody wants to admit he's Jewish. And so many people misrepresent him, scorn him, mock him, make snide comments about him without really knowing what he said or did. In this way Jesus is very, very Jewish. In his crucifixion, he joins his people in being hated without cause, in being misrepresented, in being mistreated and executed. Jesus is proof that God has not abandoned, has not forgotten his people or his promises. He has remembered his people and his treasured possession. And finally in his resurrection, Jesus is God's seal on the promise of resurrection for all people.
But for all that, we still have no neutral term for those books. All the terms we have looked at have some confessional weight: whether Old and New Testament, or Hebrew and Christian Bible. My alternatives of First Testament and New Testament, or Hebrew Bible and Worldwide Bible have their own confessional meaning, one which I am glad to stand by. If we want a value-neutral term, we might stick to "Tanakh."
Myself, I will gladly call those books the Tanakh when I am looking for a non-confessional way to refer to them. What's in a name? When the question of the name comes up, I will take the opportunity to explain why I confess what I confess.