What exactly did Jesus mean when He said, “Take up your cross and follow Me” (Matthew 16:24; Mark 8:34; Luke 9:23).
Does it mean we should carry our burdens in life or?
What Jesus means is to crucify our "old man" ie. our flesh. Paul talks about this.
We are to crucify (put off) the old man.
following is what John H. Yoder had to say about "What is our cross?" i have edited, but see no reason to rewrite it. it comes from "The way of peace in a world of war" and "Living the disarmed life".
in a sense of the word, think of the cross as a formidable spiritual weapon.
Following the example of Jesus himself, the first Christians and the writers of the New Testament were quick to see in the book of the prophet Isaiah a description of the innocent sufferings of Christ. They read there: "He was counted among evildoers. For our welfare he was chastised. Mistreated, he bore it humbly, without complaint, silent as a sheep led to the slaughter, silent as a ewe before the shearers. They did away with him unjustly though He was guilty of no violence and had not spoken one false word. " (Is 53:4-9) In all ages these words concerning the one called the "servant of the Lord" have been beloved by Christians for the portrait they paint of our crucified master. We find these same words echoing in the New Testament, not only because they are beautiful words to describe Christ and his sacrifice on behalf of sinful humanity, but also because they constitute a call to the Christian to do likewise.
There we read: "If you have done right and suffer for it, your endurance is worthwhile in the sight of God. To this you were called, because Christ suffered on your behalf, and left you an example; it is for you to follow in his steps. He committed no sin, he was guilty of no falsehood; when he suffered he uttered no threat." (1 Pe 2:20-22)
The innocent, silent uncomplaining suffering of Christ is, in this teaching of Peter, not only an act of Christ on our behalf from which we benefit; it is also an example of Christ for our instruction which we are to follow. This portrait of Christ is to be painted again on the ordinary canvas of our lives. Had not Jesus himself said that those who would follow him must deny themselves and take up their cross? What then does it mean for the Christian to bear a cross?
We meet in this world some suffering which is our own fault; we bring accidents upon ourselves by our carelessness, our punishment by our offenses. This is not "bearing a cross"; as Peter wrote, there is no merit in taking punishment for having done wrong. "What credit is it," he asks, "if when you do wrong and are beaten for it, you take it patiently?
We also sometimes suffer in ways we cannot understand, as from an unexpected or unexplained illness or catastrophe which strikes us. Such suffering the Christian can bear, trusting in God's supporting presence and learning to depend more fully and more joyfully in God's sustaining grace. Yet this is not what Jesus was talking about when he predicted suffering for his disciples.
The cross of Christ was the price of his obedience to God amid a rebellious world; it was suffering for having done right, for loving where others hated, for representing in the flesh the forgiveness and the righteousness of God among humanity, which was both less forgiving and less righteous. The cross of Christ was God's overcoming evil with good.
The cross of the Christian is then no different; it is the price of our obedience to God's love toward all others in a world ruled by hate. Such unflinching love for friend and foe alike will mean hostility and suffering for us, as it did for him.
When the apostle Paul says that "the weapons we wield are not merely human" or "not those of the world" (2 Cor 10:4), most of us, accustomed to thinking on the "merely human" level, would have expected him to say, "not human but spiritual," or "not of this world but of another world." But he says, "not merely human, but divinely potent." This is the "almighty meekness" of our reigning Lord.
When the Christian whom God has disarmed lays aside carnal weapons, it is not, in the last analysis, because they are too dangerous, but because they are too weak. The believers in Jesus as Lord direct their lives toward the day when all creation will praise, not kings and chancellors, but the Lamb that was slain as worthy to receive blessing and honor and glory and power.
No. It means we should carry ours.
Once we accept this requirement, thhe process of conversion begins; submission and change. Submission is denying self. Jesus denied himself (John 1;14), and bore our sins(Isaiah 53;5), our sins are his cross.
Our cross is to live for Jesus
To take up your cross and follow Jesus was a command to do exactly that. Had he been obeyed, there would have been twelve (or at least eleven (minus Judas)) more crosses on the hill that day. Ultimately, they did not obey...they abandoned him rather than be crucified with him.
To take it out of its context and apply it to modern America, absent of persecution, it loses the power of it. When Jesus said this, there was not a single person who heard him say it that did not know what a cross was or why it was used. There was nothing figurative or allegorical about the statement. They were very familiar with it exactly as it was said!
For application for today, if someone ever comes to you and demands that you deny Christ or die...then you need to "deny yourself, take up your cross and follow him," even to the death. If you are not willing to do this, if you would rather deny him and save your life, then you are not worthy of him (his words not mine).
Ultimately, the disciples were not worthy of him, and, in their unregenerated state, could not be his followers. Thankfully, after his death and subsequent resurrection, they were converted and became, "Christians" and stood up publically to proclaim that which previously they had been afraid to do. At that point, after conversion, they did indeed take up their cross and follow him...and almost all of them died for it (remember their conversation about who was worthy to sit next to him on the throne? He asked them, "are you able to be baptized with the baptism that I am to be baptized with?" referring to his death. They said yes, but they proved that they were not.
The references of Paul telling a Christian to put off the old man, if you look at the context, are all past tense (as something that has already happened). The Ephesians 4:22 verse is often taken out of its context to say it is something we need to do. But it reads like this, beginning 4:20:
"But ye have not so learned Christ; If so be that ye have heard him, and have been taught by him, as the truth is in Jesus: (20-21)"
What did they learn as truth? We don't have to wonder Verses 22-24 tell us, they say:
That ye put off concerning the former conversation the old man, which is corrupt according to the deceitful lusts; And be renewed in the spirit of your mind; And that ye put on the new man, which after God is created in righteousness and true holiness.
It is past tense language. This is not an admonition to do anything, it is a reminder of what has already been done. He goes on in the passage to talk about the fact that we have the power to live a certain way based on the fact that the old man is dead already.