Having a desire to let our light shine and to share our faith are positive goals! In order to answer this specific question, though, we must also consider the background of the cross, the New Testament record and Jesus' teaching about how to display our Christianity.
One of the first questions we might ask is, Who decided that the cross was to be the sign of Christianity? This tradition of wearing a cross does not come from the Bible or the practices of the New Testament Church. Though there are at least seven different types of crosses, we are not even certain that Jesus was crucified on a cross-like apparatus. Though crucifixion on a cross was common at the time, the Bible allows that Jesus may have been put to death on an upright pole (rendered "tree" in 1 Peter 2:24 1 Peter 2:24Who his own self bore our sins in his own body on the tree, that we, being dead to sins, should live to righteousness: by whose stripes you were healed.
American King James Version×from the Greek word stauros, which primarily means an upright stake).
A study of history shows that the cross symbol predates Christianity. According to author Ralph Woodrow, "Centuries before the Christian era, the cross was honored as a religious symbol by the people of Babylon. It is seen on their oldest monuments. Historians say that it was a symbol associated with Tammuz" (Babylonian Mystery Religion, p. 51). From Babylon, the cross spread to other nations and was associated with paganism long before Jesus' crucifixion in A.D. 31.
Woodrow further explains, "It was not until Christianity began to be paganized that the cross came to be thought of as a Christian symbol. It was in 431 A.D. that crosses in churches and chambers were introduced, while the use of crosses on steeples did not come until about 586 A.D." (p. 50).
While most people today connect the cross with Christianity rather than paganism, we must also ask if the cross is something to be worshipped or honored. While the apostles preached "the cross [stauros]" as part of the history of Christ's ministry for our sakes (1 Corinthians 1:17-18 1 Corinthians 1:17-18  For Christ sent me not to baptize, but to preach the gospel: not with wisdom of words, lest the cross of Christ should be made of none effect.
 For the preaching of the cross is to them that perish foolishness; but to us which are saved it is the power of God.
American King James Version×), it was not something they idolized. It was a shameful instrument of death (Hebrews 12:2 Hebrews 12:2Looking to Jesus the author and finisher of our faith; who for the joy that was set before him endured the cross, despising the shame, and is set down at the right hand of the throne of God.
American King James Version×). In His crucifixion, Jesus took on Himself our shameful sins. Having our sins forgiven is a wonderful blessing, but there is no need to glorify the instrument used.
Finally, consider what the Bible teaches about wearing any religious symbol. Under the Old Covenant that God made with ancient Israel, God instructed them to wear reminders of their faith upon their hands (Deuteronomy 6:8 Deuteronomy 6:8And you shall bind them for a sign on your hand, and they shall be as frontlets between your eyes.
American King James Version×; Deuteronomy 11:18 Deuteronomy 11:18Therefore shall you lay up these my words in your heart and in your soul, and bind them for a sign on your hand, that they may be as frontlets between your eyes.
American King James Version×). In fulfillment of this command, phylacteries, small leather boxes containing scriptural passages, were traditionally worn by Jewish men during their morning weekday prayers. Many did this to appear righteous to others (Matthew 23:5 Matthew 23:5But all their works they do for to be seen of men: they make broad their phylacteries, and enlarge the borders of their garments,
American King James Version×).
During His New Testament ministry, Jesus taught His followers to display their spirituality through their actions and deeds (Matthew 5:16 Matthew 5:16Let your light so shine before men, that they may see your good works, and glorify your Father which is in heaven.
American King James Version×). Under the New Covenant, ushered in by Christ, God's laws are to be written on our hearts—that is, in our minds (Hebrews 8:10 Hebrews 8:10For this is the covenant that I will make with the house of Israel after those days, said the Lord; I will put my laws into their mind, and write them in their hearts: and I will be to them a God, and they shall be to me a people:
American King James Version×; Hebrews 10:16 Hebrews 10:16This is the covenant that I will make with them after those days, said the Lord, I will put my laws into their hearts, and in their minds will I write them;
American King James Version×). People who truly practice the Christianity of the Bible stand out as beacons of light in a spiritually darkened society because of the way they live. They have no need to wear external signs like a cross to identify themselves as Christian.
If you would like to learn more about the history of the cross and how this pagan symbol entered Christianity, read chapters 6 and 7 of Ralph Woodrow's book, Babylon Mystery Religion. These two chapters are respectively titled "Is the Cross a Christian Symbol?" and "Constantine and the Cross."