The dispensational premillennialists argue that Israel is God's chosen people and all the promises and predictions of the Old Testament only apply to the Jewish nation, not to Christians.
They see the present "age of grace" or "church age" as a unique historical period in which God deals with Jews and Gentiles alike through the death and resurrection of Jesus. They like to say that the church age is a "parenthesis" between two exclusively Jewish ages - the age of the Law (from Moses to Jesus) and the Millennial kingdom (which follows the secret rapture which removes all true Christians from the earth).
The key to dispensationalism is that God deals with humans by different standards and offers different criteria for judgment in each progressive period (dispensation) of human history. According to this view, the challenge of reading the Bible is to "rightly divide the word of truth" - specifically, to distinguish those parts of the Bible that focus on Jews (which have nothing to do with Christianity) and those parts which focus on the Christian church (which have no application to national Israel).
So, it is not correct to say that for the dispensationalist, the old covenant has failed and been replaced with a new covenant. Rather dispensationalists would argue that the old covenant (testament) speaks of Israel only, while the new covenant (testament) speaks only of the Christian church (except for some selected sections of the New Testament - especially the Book of Revelation - that they understand to deal explicitly with Israel). Dispensationalism is not supersessionist - a replacement theology that says that Christianity has replaced Judaism. Rather, Christianity is a momentary "blip" in God's larger plan of working through Israel. According to this view, with the removal of the church via the rapture, God will get back to his original way of "doing business" working in and through the Jewish people.
Personally, I cannot accept this position because both Jesus and Paul seem to be very clear that the new covenant prophesied in Jeremiah 31 is fulfilled in the New Testament followers of Jesus - both Jew and Gentile. (See especially the eucharistic words of Jesus, "This is the new covenant in my blood", along with Paul's extensive argument about Israel's continued place in the people of God in Romans 9-11.)
Jesus and Paul taught that the end of the current age has come and the powers of sin, the Satan, and death have been defeated. (The resurrection of Jesus is the "first fruit" of the dawning new age and a sure and certain sign that the powers have been defeated.) The "age to come" will still include Israel as the "people of God." But now at the end of time, the promises of the Old Testament prophets - that the law will flow forth from Zion, that light will shine on the Gentiles, and that they too will be included along with Israel in the "people of God" without converting to Judaism first - will at last be fulfilled.