It might be worthwhile to take a look at the passage from Matthew that introduces the idea of "taking up your cross":
From that time on, Jesus began to show his disciples that he must go to Jerusalem and suffer greatly from the elders, the chief priests, and the scribes, and be killed and on the third day be raised.
Then Peter took him aside and began to rebuke him, "God forbid, Lord! No such thing shall ever happen to you."
He turned and said to Peter, "Get behind me, Satan! You are an obstacle to me. You are thinking not as God does, but as human beings do."
Then Jesus said to his disciples, "Whoever wishes to come after me must deny himself, take up his cross, and follow me. For whoever wishes to save his life will lose it, but whoever loses his life for my sake will find it."
I shall now demonstrate my mastery of the obvious.
The disciples are here taught three lessons: that Jesus is the Messiah; that the Messiah must suffer; and that those who would follow the Messiah must suffer likewise.
Only after His disciples confess that He is the Messiah does Jesus begins to teach them that the Messiah must suffer and die. Led by Peter, the disciples first learn who Jesus is (more or less), and only then does He go on to say what that truly implies. That He must teach them step by step is vividly illustrated by Peter's reaction to Lesson Two.
In this passage, Lesson Three follows hard on Jesus' grading of Peter's comprehension of Lesson Two. Note the repetition of "must" in these lessons: "that he must go to Jerusalem and suffer greatly," "Whoever wishes to come after me must deny himself." To be a student is to imitate the master. Perhaps that thought added a little extra to Peter's emphatic, "God forbid, Lord!"
Okay, now I come to the crux (man, I slay me) of the matter:
"Whoever wishes to come after me must deny himself, take up his cross, and follow me."
What Jesus describes is a three-step process, and I would suggest that understanding the second step requires understanding the first step.
The NAB note on this says that "to deny oneself is to disown oneself as the center of one's existence." The Catena Aurea quotes St. Gregory as writing, "He denies himself whosoever is changed for the better, and begins to be what he was not, and ceases to be what he was... He also denies himself, who having trode under foot the risings of pride, shews himself in the eyes of God to be estranged from himself." St. John Chrysostom is more expansive:
He that disowns another, whether a brother, or a servant, or whosoever it be, he may see him beaten, or suffering aught else, and neither succours nor befriends him; thus it is He would have us deny our body, and whether it be beaten or addicted in any other way, not to spare it.
To deny yourself, then, is to change, if not completely sever, your current relationship with yourself. Broadly (and figuratively) speaking, the change involves taking your will -- the satisfying of which is, for those turned away from God, the greatest good -- and trampling it underfoot.