The history behind the kiss of peace is interesting. I won't go into detail because I would need to dig out a few books, but basically it is a part of all traditional liturgical rites and was considered an important, if not essential, part of the Liturgy and of the liturgical preparation for receiving Communion. Technically, that hasn't changed.
I'm not sure exactly what form it took in the beginning, but presumably it included an actual kiss, perhaps two or three if such customs go way back, and some sort of embrace. One would exchange the kiss with everyone in the church (contemporary examples of this practice can be seen in the rite of forgiveness before Lent and the exchange of the Paschal kiss). Not only does that take a considerable amount of time, but it was conducive to certain abuses. For example, adolescents hitting on each other, some getting a "longer" or more earnest kiss than others, people using it as a "break" from the service and running off into town for no good reason, etc. If I'm not mistaken, Fr Taft discusses this in his book Through Their Own Eyes: Liturgy as the Byzantines Saw It, but I seem to have misplaced my copy. As I remember, he includes a quote from a sermon of St John Chrysostom where he just rips into everyone and all but condemns them for doing this sort of stuff during the Liturgy.
So the kiss began to be restricted, and this occurred in various ways in different places. One common method was to restrict the kiss to certain "classes". The priests would exchange the kiss among themselves, and the deacons among themselves, and the men among themselves, the women among themselves, and so on. Eventually, this became restricted even further until we get to the received practice in the traditional Roman and Byzantine rites: the kiss is exchanged only among the clergy. Remnants of the past can be seen in various rituals or textual hints, but you have to know what you're looking for: chances are the average worshiper would have a completely different understanding of what those meant, and "greet one another with a holy kiss" wouldn't be one of them.
Another method of curbing abuses was to change the manner of the kiss so that it wasn't really an actual kiss, but a stylised kiss. This is the practice which has been received by all the Oriental Orthodox traditions, which retain the kiss of peace in the Liturgy to this day. So, for example, Copts and Syrians (and Churches of these traditions) clasp hands: the giver holds his hands in a "praying hands" fashion but leaves them a little open, while the receiver holds his hands in the same way, and each clasps the right hand of the other. This may be preceded and/or followed by each person touching their own lips as a way of giving and receiving the kiss. Armenians, on the other hand, replicate the double kiss with an exchange of greetings, but there's no actual kiss. It looks a little like two people are slightly bowing to each other's shoulders. While Copts simply exchange the peace among their neighbours when the command is given, the Syrians and Armenians receive it from the altar: the priest prays to receive the peace from God, offers the liturgical greeting of peace, passes the peace to the deacon, who then may exchange it with the other deacons before bringing it to the people, who will pass it among each other and on towards the back of the church.