I don't think I remember seeing the Kiss of Peace at the Russian OCA church I went to two weeks ago either.
The kiss of peace is generally limited to the clergy in EO churches, except for a limited number of parishes (mainly in America) where they have made efforts to reinstate it.
It's interesting that the kiss in its original form has disappeared from all churches. Even in churches where it has been retained, it has turned into a hand-shake or something similar.
Although the kiss of peace as practiced today in the Coptic tradition might loosely be classified as a "handshake" of sorts, its purpose is far from some sort of social "meet and greet" interlude. I am furthermore not entirely certain that the use of the hands in contemporary Coptic practice reflects some sort of compromise between a need to retain the general ritual and some modern concern to avoid direct kissing; for two reasons:
1) As a matter of common sense, there is no risk of the rite being improperly practiced given that males and females practice it strictly amongst their own gender. It is common in Egyptian culture for males to greet other males socially with a kiss on each cheek, and likewise for females.
2) The use of the hands appears to effectively convey a particular moral notion central to the general purpose of the kiss of peace. Before I elaborate on this point I think it'd be helpful to first detail the motions of the kiss of peace in Coptic practice:
Typically one turns to those in the immediate vicinity and extends both hands outwards perpendicularly and spaced a little. The respondent does likewise, and as he/she extends his/her own hands towards those of the other, he/she inserts one of his/her hands in between the two hands of the other (and thereby receives one of the other's hands in between his/her own). Both parties than clasp their hands tightly together, and then pull their hands inwards and away, until the enclasped hand of other is released. They then bring the fingertips of their clasped hands towards their lips whereupon they kiss them.
The symbolism of such a gesture was explained by a certain priest as follows: in clasping both sides of the other's hands with one's own two hands, one signifies an holistic embrace of the other, which is to say that one expresses acceptance of them along with their good qualities (symbolised by the smooth side of the hand i.e. the palm side) and their negative qualities (symbolised by the rougher side of the hand). In short, it is an expression of unconditional love.
In its liturgical context the kiss of peace signifies the need for us to be reconciled amongst ourselves before we can benefit from Christ's gift of reconciling us to God. If we are not willing to engage in loving fellowship with others because of their faults and weaknesses, then how can we expect Christ to lovingly indwell us, with all of our own faults and weaknesses, by partaking of His Holy Body and Precious Blood?