Our Bodies Will Live for Eternity

It's become popular within recent history for Christians to think we'll live on for eternity as disembodied spirits or souls. The idea is that our physical bodies will rot or be destroyed and we'll be free from them in heaven.

This is not the early Christian belief, and it was never the Christian belief until modern times. This idea isn't much different from the ancient Greek belief that the disembodied shades of the dead dwell in Hades, and if that was the resurrection Paul was preaching on Mars Hill his listeners would have been utterly bored rather than offended by it.

Alexander Campbell, the leading Restoration Movement thinker, advocated the original Christian idea of bodily resurrection and excoriated the false modern idea in his 1835 book The Christian System:
"It was great to create man in the image of God - greater to redeem his soul from general corruption; but greatest of all to give his mortal frame incorruptible and immortal vigor. The power displayed in the giving to the dead body of the Son of God incorruptible glory and endless life is set forth by the Apostle Paul..."
"To raise a dead body to life again is not set forth as more glorious than by a touch to give new vigor to the palsied arm, to impart sight to the blind, or hearing to the deaf; but to give that raised body the deathless vigor of incorruptibility, to renovate and transform it in all its parts, and to make every spirit feel that it reanimates its own body, that it is as insusceptible of decay, as immortal as the Father of eternity, is a thought overwhelming to every mind...."
"Immortality, in the sacred writings, is never applied to the spirit of man. It is not the doctrine of Plato which the resurrection of Jesus proposes. It is the immortality of the body of which the resurrection is a proof and pledge. This was never developed till he became the first-born from the dead, and in the human body entered heaven. Jesus was not a spirit when he returned to God. He is not made the Head of the New Creation as a Spirit, but as the Son of Man... we shall be made like him, we shall see him as he is.... by the word of his grace he reanimates the soul of man; and by the word of his power he will again form our bodies anew, and reunite the spirit and the body in the bonds of an incorruptible and everlasting union. Then shall death 'be swallowed up forever.'"
Read 1 Corinthians 15:12-13 if you would like more context about this discussion.