What does ‘take up your cross’ mean?

Then said Jesus unto his disciples, If any man will come after me, let him deny himself, and take up his cross, and follow me (Matthew 16:24).

And when he had called the people unto him with his disciples also, he said unto them, Whosoever will come after me, let him deny himself, and take up his cross, and follow me (Mark 8:34).

And he said to them all, If any man will come after me, let him deny himself, and take up his cross daily, and follow me (Luke 9:23).

Meaning of ‘take up the cross’

And he that taketh not his cross, etc. When persons were condemned to be crucified, a part of the sentence was, that they should carry the cross on which they were to die to the place of execution. Thus Christ carried his, till he fainted from fatigue and exhaustion. The cross was usually composed of two rough beams of wood nailed together.  It was an instrument of death. To carry it was burdensome, was disgraceful, was trying to the feelings, was an addition to the punishment. So, to carry the cross is a figurative expression, denoting that we must endure whatever is burdensome, or trying, or considered as disgraceful, in following Christ. Reference: Albert Barnes Commentary  

To take up the cross means that we should be ready to suffer hardships and afflictions for the sake of Christ or for Christianity. Reference: British Family Bible

To take up our cross and follow Christ is to deny ourselves or to mortify our flesh and to renounce the world, and all its pleasures, connections, and concerns, to forego everything that is cheerful and agreeable to our nature, and to consume our whole life in abstinence and austerity. Reference: British Family Bible

Taking up one’s cross is being prepared for trials in general for Christ’s sake, or it is the preparedness to go forth even to crucifixion for Christ’s sake. Reference: Jamieson Fausset Brown

Meaning of ‘He that taketh not his cross’

He that taketh not his cross – here Jesus meant to say, “He who is not ready, after my example, to suffer death in the cause of my religion, is not worthy of me, does not deserve to be called my disciple”. Reference: Adam Clarke

He that taketh not his cross means that he that does not willingly, and cheerfully, and patiently bear and undergo those trials, and afflictions, and persecutions, which God in the way of his providence shall lay upon him, and bring him into, for my sake and my gospel, is not worthy of the name or reward of my disciples. Reference: Matthew Poole’s Commentary

He that taketh not his cross means he that is not prepared to endure the greatest ignominy and suffering which may be connected with following me; he that is not prepared even to die in a most ignominious and painful manner, rather than renounce me, is not in a fit condition to be my disciple. Reference: Ribley 4 Gospels