Often when referring to the resurrection, Christians will speak of receiving their "new" body. That way of speaking is not necessarily wrong if the meaning is that our current bodies will be renewed so that they are "as good as (or, better than) new." But we should not think of the resurrection as the reception of a new body in the sense that we are given a different body disconnected from the body we had on earth.
Instead, the Bible teaches that the resurrection is a transformation of the same bodies we had on earth. As humans, we are not just spiritual, but physical. Our bodies are a very important part of our identity--they are part of who we are. Therefore, if we deny that we are raised with the same bodies we had on earth, we are denying a significant part of our identity. At the same time, if we deny that our resurrected bodies are transformed, we are left with the depressing idea that we will forever be subject to the weaknesses we now have, such as sickness, fatigue, etc. As Piper has said: "The old body will become a new body. But it will still be your body. There will be continuity. God is able to do what we cannot imagine. The resurrection is not described in terms of a totally new creation but in terms of a change of the old creation" (Future Grace, 372).
We will have the same bodies
There are many Scriptural reasons for believing that we will be raised with the same body that died. First, Christ was raised in the same body He had before He died. We know this because the tomb was empty (Luke 24:1-6) and because His resurrected body retained scars from the crucifixion (John 20:25, 27). Since Christ's resurrection is the pattern that our resurrection will follow (Philippians 3:20-21; 1 Cor. 15:49), then we will also be raised with the same body.
Second, this is also evident from the very meaning of the term "resurrection of the dead" (1 Corinthians 15:13, etc.). The phrase means: that which is dead (namely, our body) is made alive. If the same body that died is not the body that was raised, Paul could not call it the "resurrection of the dead." It would not be a resurrection at all.
Third, the phrase "the dead will be raised" (1 Cor. 15:52) also communicates this. John Piper comments on this verse that, "If God meant to start all over with no continuity between the body I have now and the one I will have, why would Paul say 'the dead will be raised'? Why would he not say, 'the dead will not be raised (since they are decomposed and their molecules are scattered into plants and animals for a thousand miles) and so God will start from scratch'? He did not say that, because it is not true" (Future Grace, 372).
Fourth, Philippians 3:20-1 says that our earthly body is transformed into conformity with Christ's body in the resurrection, not that God creates a new body from scratch: "For our citizenship is in heaven, from which also we eagerly wait for a Savior, the Lord Jesus Christ; who will transform the body of our humble state into conformity with the body of His glory, by the exertion of the power that He has even to subject all things to Himself."
Fifth, Jesus speaks of the resurrection as involving the coming forth out of tombs, which strongly indicates that the resurrection is the reanimation of the body that had been lied to rest originally: "An hour is coming, in which all who are in the tombs will hear His voice, and will come forth; those who did the good deeds to a resurrection of life, those who committed the evil deeds to a resurrection of judgment" (John 5:28-29).
Sixth, Paul's statement "it is sown a perishable body, it is raised an imperishable body" (1 Corinthians 15:42) establishes that there is a continuity between our current body and our resurrected body, for it is the same "it" in both cases.
Seventh, verse 53 indicates that the same body we have now (which is mortal), will become immortal: "For this perishable must put on the imperishable, and this mortal must put on immortality."
We will have transformed bodies
In 1 Corinthians 15:35-37, it may appear as if Paul is teaching that we are raised with a different body than which we had on earth: "...what you sow is not made alive unless it dies. And what you sow, you do not sow the body that shall be, but mere grain." But upon examining the whole context, we see that Paul is not denying that it will be the same body. Instead, he is affirming that in the resurrection our bodies will be made better than the state they are now in.
In fact, this passage teaches a continuity between our bodies now and in the resurrected state by using the analogy from agriculture. Paul compares the resurrection of the body to the growth of a plant from a seed. The plant that results is definitely much better than the seed, just as our resurrection bodies will be better than those we have now. But there is also a real continuity between the seed and the plant, for they are the same organism. The same seed that was sown becomes the plant that grows. Likewise, the same body we have now becomes our resurrected body. But just as the plant is a result of the seed being transformed into something with better capacities and qualities, so also in the resurrection our bodies will receive better qualities and capacities. Thus, when Paul says that we do not yet have the body that shall be, he means that our current bodies are not yet in their glorified and improved state (see verses 42-44). They are not as they will be.
Paul also affirms that the resurrection involves the transformation of our current bodies in 1 Corinthians 15:51-52. "Behold, I tell you a mystery; we shall not all sleep, but we shall all be changed, in a moment, in the twinkling of an eye, at the last trumpet; for the trumpet will sound, and the dead will be raised imperishable, and we shall be changed." John Piper comments: "He said two things: the dead will be raised (that teaches continuity); and the dead will be changed (they will be made imperishable and immortal)" (Future Grace, 372).
In what sense will our bodies be transformed? Paul tells us in verse 42-44. He says that our current bodies are weak, perishable, unglorified, and natural. But in the resurrection state they will be powerful, imperishable, glorious, and spiritual. Our bodies will be powerful--they will not be subject to stress or fatigue or weakness. Our bodies will be imperishable--they will not get sick, die, age, or become injured. Our bodies will be spiritual--they will be fully oriented to and filled with the Holy Spirit. And our bodies will be glorious. Wayne Grudem comments on the wonder of this truth:
Because the word 'glory' is so frequently used in Scripture of the bright shining radiance that surrounds the presence of God himself, this term suggests that there will also be a kind of brightness or radiance surrounding our bodies that will be an appropriate outward evidence of the position of exaltation and rule over all creation that God has given us. This is also suggested in Matthew 13:43, where Jesus says, 'Then the righteous will shine like the sun in the kingdom of their Father.' Similarly, we read in Daniel's vision, 'And those who are wise shall shine like the brightness of the firmament; and those who turn many to righteousness, like the stars forever and ever' (Daniel 12:3). (Wayne Grudem, Systematic Theology, 833).
John Piper, Future Grace, chapter 30, "The Rebirth of Creation"
John Piper, "Our Hope: The Redemption of Our Bodies"
Wayne Grudem, Systematic Theology, chapter 42, "Glorification"
John Murray, Redemption: Accomplished and Applied, "Glorification"