According to the excellent compendium on the Typikon "Σύστημα Τυπικού" by Protopresvyter Father Konstantinos Papagiannis the Koinonikon (page 90, sections 128 and 129, Apostliki Diakonia tis Ekklhsias tis Ellados, First Edition 2006; I am translating and paraphrasing with some explanatory notes from this point onwards):
"The Koinonikon refers to the hymn chanted at and during the period of the Holy Communion in the service of the Holy Liturgy. In earlier times, the koinonikon consisted of an entire psalm whereupon a characteristic efymnion [verse] ending with Alleluiah would follow each of the psalmic verses. In this manner, the chanting of the koinonikon would cover the entire time in which the faithful would receive communion. With time, however, the psalmic verses fell into disuse and what remained was the characteristic efymnion that was chanted in a slow manner once and only at the time where the clergy received communion. The invocation "Tou Deipnou sou tou mystikou" was then used at the time at which the faithful received communion."
Papagiannis footnotes this section (footnote 118) with the following observation and opinion (again translated and paraphrased):
"Today, many chanters, in violation of formality, have replaced the koinonika with verses of Polyeleoi or other quasi-psalmic or psalm-inspired verses (e.g. Που πορευθω απο του πνευματος σου....Κυριε ο Κυριος ημων) which has no relation to the context of the celebration (the day) or the feast. It is more correct and appropriate to return to the old practice, to chant the koinonikon in a brief manner, repeating the specific efymnion verse with the Alleluiah after each of the psalmic verses or those psalmic verses of relevance to the feast and to continue the chanting of the psalm in this manner throughout the period at which the faithful receive communion".
I append a scan of the page for those who wish to view the original Greek text.
SOME THOUGHTS, especially for those outside of Greece (and maybe some inside Greece!).
The comments by Fr. Papagiannis are indeed correct and this is what we should all strive for.
However at the same time, let us consider the following:
1) The Polyeleos is omitted at the orthros for certain feasts in almost all churches in Greece and to my knowledge in all the churches outside Greece
2) The Canon (especially the Canon of the celebrating saint in the Minaion, or of the day in the Triodion) is omitted in its entirety in almost all churches outside of Greece and maybe even some in Greece
3) Most faithful come to church at the end of the Orthros and at the Liturgy
Given the high didactic value, especially of the Canon, perhaps consideration could be given to permit chanting of as many troparia of the odes as possible during the communion of the faithful, keeping in place the koinonikon prior to "Meta fovou Theou".
A lot worse is happening now (kratimata-some self-composed that are outside any tradition, unrelated chants including the Agni Parthene where an edict by the Ecumenical Patriarchate has frowned on its use and so on).
I am not taking a position for or against the polyeleoi or anything else, but I share these thoughts for discussion. I am "guilty" of polyeleos from time to time at the communion of the faithful, although the choice is in context of the day (or the feast).
Ultimately, the Ecumenical Patriarchate will have to sit and seriously consider and reconsider what is appropriate. Until then, it is safest and historically prudent to seriously adopt what Fr. Papagiannis (and tradition) writes.
NG