New Testament References
Paul, in his chronologically first New Testament writing, refers to something called a holy kiss. This forgotten practice was common among first century Christians.
Greet all the brethren with a holy kiss (1Thessalonians 5:26, HBFV throughout).
Paul also refers to a holy kiss in his closings for the book of Romans (Romans 16:16) and his two letters to Corinth (1Corinthians 16:20, 2Corinthians 13:12). The Apostle Peter also references this act between brethren using language similar to Paul's.
Greet one another with a kiss of love. Peace be with all of you who are in Christ Jesus. Amen (1Peter 5:14).
Acknowledged by Jesus
Jesus, during his ministry, acknowledged the existence and importance of a kiss as a common form of greeting.
A Pharisee, in the summer of 28 A.D., invited Jesus to dine with him. After the Lord entered his home a woman (likely a prostitute) came in and began to kiss and anoint his feet while crying.
The Pharisee, who knows the woman, concludes in his heart that if Jesus were a prophet he would not let such a sinner touch him. The touch of such a sinful person, based on the Pharisee's beliefs, rendered the Lord ceremonially unclean.
Part of Jesus' correction of the Pharisees' self-righteous attitude included mentioning the leader's lack of offering courtesies such as a kiss.
And after turning to the woman, He (Jesus) said to Simon, "Do you see this woman? I came into your house, and you did not provide any water to wash My feet; but she has washed My feet with her tears and wiped them with the hairs of her head.
"You did not give Me a kiss; but she, from the time I came in, has not ceased to ardently kiss My feet" (Luke 7:44 - 45, HBFV).
It was common, in Jesus' day, for a host to greet his guests with a kiss and then offer them water to wash their feet. The host would also offer a little oil to anoint the heads of his guests (Luke 7:46).
The use of a non-sexual kiss to greet people was so common that it was used by Judas Iscariot to identify Jesus to those who wanted him dead (Matthew 26:48, Mark 14:44, Luke 22:47 - 48).
More Than a Greeting
The holy kiss mentioned by both Paul and Peter is far more than a simple greeting between two people. For Christians it was (and is) an outward sign, an external acknowledgment of the spiritual connection shared by all those who have God's spirit.
The Bible teaches that Christians have a spiritual familial link to God as Father and Jesus Christ as our elder brother (Romans 8:29, Ephesians 3:14 - 15, Hebrews 2:11). The eternal relationship between believers as spiritual brothers and sisters is the reason why Christians are collectively referred to as brethren more than 140 times in the New Testament.
Should We Do It?
It certainly would not be wrong for Christians to greet each other today with a holy kiss on the cheek. That said, modern displays of greeting and affection such as hugs, a pat on the back, or a warm friendly handshake are also acceptable alternatives.