Question: "Is it necessary for an Orthodox Christian to always wear a Cross?"
The practice of wearing a cross is very ancient, and it is done for two reasons: 1). It is a confession of faith that we are Christians; and 2). The Cross is a weapon against evil.
In Russia, to say that someone "took off his Cross" is another way of saying that they have renounced their Christian Faith. One of the most popularly venerated saints of recent times is the soldier Evgeny Rodionov, who was captured by the Muslim Chechens during the First Chechen. He was beaten and tortured, and finally beheaded because he refused to remove his Cross.
There are some good reasons why someone might need to remove their Cross temporarily, perhaps because of some work safety issues, but aside from such exceptional circumstances, one should wear their Cross at all times.
For laymen, the wearing of the Cross is worn next to the skin, and so usually is not visible to others, but is a constant reminder to one's self of their faith. However, it can at times be seen by others, and so is also a testimony to others, as it was most clearly in the case of the New Martyr Evgeny.
On the back of most Orthodox Cross, you will find one of two texts. The most common one is "Спаси и Сохрани," which means "Save and Protect". One also sees this prayer, though usually not the entire prayer, unless the Cross is fairly large:
"Да воскреснет Бог, и расточатся врази Его, и да бежат от лица Его ненавидящии Его. Яко исчезает дым, да исчезнут; яко тает воск от лица огня, тако да погибнут беси от лица любящих Бога и знаменующихся крестным знамением, и в веселии глаголющих: радуйся, Пречестный и Животворящий Кресте Господень, прогоняяй бесы силою на тебе пропятаго Господа нашего Иисуса Христа, во ад сшедшаго и поправшаго силу диаволю, и даровавшаго нам тебе Крест Свой Честный на прогнание всякаго супостата. О, Пречестный и Животворящий Кресте Господень! Помогай ми со Святою Госпожею Девою Богородицею и со всеми святыми во веки. Аминь."
"Let God arise and let His enemies be scattered, and let them that hate Him flee from before His face. As smoke vanisheth, so let them vanish; as wax melteth before the fire, so let the demons perish from the presence of them that love God and who sign themselves with the sign of the Cross and say in gladness: Rejoice, most precious and life-giving Cross of the Lord, for Thou drivest away the demons by the power of our Lord Jesus Christ Who was crucified on thee, Who went down to hades and trampled on the power of the devil, and gave us thee, His precious Cross, for the driving away of every adversary. O most precious and life-giving Cross of the Lord, help me together with the holy Lady Virgin Theotokos, and with all the saints, unto the ages. Amen."
This prayer begins with the words of Psalm 67, but go on to interpret that prayer as a prayer against our invisible enemies, the demons, who are defeated by the power of the Cross.
Wearing a Cross is not a lucky charm that would be of any benefit to an unbeliever, nor is it a guarantee that even a pious Orthodox Christian will experience no physical harm while wearing it; but for those who do wear it with faith in the power of the Cross of Christ, it is of great spiritual comfort and benefit.
For more information see:
What the Cross Means for Christians, by Professor I.M. Andreyev
The Wearing of Christian Baptismal Crosses, by Igumen Philip (Ryabykh) and Igor Ponkin
Answers to Practical Questions About The Orthodox Way of Life, No. 1 - On Wearing the Cross