The history of the “holy kiss” originates in the New Testament, where in four different letters from Paul to different churches, the early Christians are instructed to “greet one another with a holy kiss” (Romans 16:16, 1 Corinthians 16:20, 2 Corinthians 13:12, 1 Thessalonians 5:26). Some churches (not all necessarily Mennonite) take these passages to heart, and greet one another with a kiss on the cheek, much as anyone else would greet another person with a hug or handshake. This tends to occur in the more conservative Mennonite churches, and was more prevalent in the past. In the conservative Mennonite and Amish churches where this practice continues, it is strictly understood that the holy kiss is shared only between persons of the same sex.
Most modern Mennonites read these passages in a cultural context. While they still greet one another warmly and offer blessings to one another in various ways, they usually don’t actually kiss.