The English translation hides two important pieces of wordplay in the original Greek.
The translation of this verse as "taking up a cross" was only possible after Christ's death. During his lifetime, people would have heard this as "lifting up a walking stick" or "pulling up a stake," as in a tent stake, which was also used as a walking stick. While people were killed by staking them up, no one at the time would have heard this verse as referring to a death stake or carrying one before Jesus's death. Tent stakes, house posts, and walking sticks were much more common.
The first relates to the previous verse where Jesus says, "Get behind me, adversary." The two verses seem disconnected in English translation, but in Greek, one logically follows the other. Both verses use the same Greek phrase (opiso mou) translated first as "behind me" and then as "after me." In the previous verse, "Get behind me" seems like a rebuke but in light of this verse, it is more of a request for support. It is exactly like English when we ask people to support us by "getting behind us." In this verse, Jesus completes the request. How do support Christ? By denying ourselves and following him.
Christ's Words in Matthew as a Guide to 40 Days of Prayer.