. Why do the Eastern Orthodox Churches not do this ceremony??? Thank you.
The Eastern Orthodox Churches do include the Kiss of Peace in the Liturgy, however, it is now usually only done by the concelebratating clergy at the Altar. It is most often seen these days in large monasteries and cathedrals where more than one celebrant is usually present. The Kiss of Peace signifies not only reconciliation, but unity of faith. The practice was stopped among the laity in the Eastern Orthodox Churches, (along with the Agape) possibly due to abuses, however it has always been an integral part of the Liturgy since Apostolic times. See: Romans 16:16, 1 Corinthians 16:20, 2 Corinthians 13:12, 1 Thessalonians 5:26, 1 Peter 5:14.
In a similar way, the Rite of Artoclasia at Vespers is a remnant of the Agape (love Feast) of Apostolic times.