Were there any Christian groups that existed during the time of the first four ecumenical councils and that were not represented at them either because they chose not to attend or they were not invited or, perhaps, were not even known of?
The Nestorians are a good example for the latter two councils. There were a few attendees from the non-Greek east, groups that would later be called Nestorians. There was at least two, Jacob of Nisibis and another bishop named John. The eastern church "officially" accepted Nicea in 410 at the council of Isaac though the Nestorian schism happened soon after. But I guess nobody doubts the existence of the now-called Nestorian church during the time. Some of their writings survive and I'd recommend a reading of Moffett's "Christianity in Asia".
If I recall correctly, it is still too early for the Indian Christians to be associated very closely with the churches later called "Nestorian". Perhaps they are as old as Thomas. One tradition says it was Bartholomew rather than Thomas. It is reported that when Pantaenus went to India in the second century from Egypt he found Christians there. So there is probably an old tradition there, but as far as I know none of theme made it to the ecumenical councils.
The Chinese churches probably did not yet exist at the time of the first councils.
Keep in mind that groups east of the Roman Empire might have had troubles at times traveling to councils in the west. The Romans and Persians were on and off again at war, which would have hurt the communication lines a bit I'm sure, even if travel wasn't completely stopped.
If you go further west, there are traditions of Christianity reaching England in the second century, though as far as I know it didn't really thrive till much later (but I could be wrong on that). But if there are groups that far west, it is less likely that they would have made it to an ecumenical council.
We know the Marcionites existed, even after the death of Marcion. As far as I know, they had no presence at the councils and none of their writings survive.
Some lump them the Marcionites in with the Christian gnostics (and not all gnostics would have attached themselves to Christianity), but it might just be best to say there were a number of different types of gnostic groups. We have evidence of various gnostic groups because of polemics against them by the early fathers. If you haven't read Irenaeus, you can see a variety of gnostic groups attacked there. Since the discoveries of the Nag Hammadi documents we also have evidence of gnostics in their own words. You can also see some material on the gnostics in the early Christian historians.
There are also the Ebionites, who were a group of law-following Christians that some would say are akin to what people think of when they think "Judaizer". They were still around to a certain degree during the councils but as far as I know did not attend and no writings survive.
The Arians were around during the councils but they attended...and were ultimately condemned. So they don't really fit your question, but I figured I would mention them anyway.
So if you want to read for yourself from the groups that were eventually considered heretics by the orthodox, you are going to have trouble finding primary source texts to read. Most of what we have are from their opponents, which we can't assume always portray their opponents fairly, even if some (like myself) think they were right in deeming the ideas unorthodox (well...at least I think they were mostly right)
This may not exactly answer your question, but perhaps the most significant presence at the first few councils which was eventually thrown out was the Nestorian church.
We have lots of evidence that they existed, and in fact they were a major influence in the "Church of the East" until about 622 AD. Syria, Persia, and much of the East was Nestorian, prior to the rise of Islam. There is evidence, that there was a significant Nestorian presence in China as late as the 9th Century, as evidenced by a Steele rediscovered in 1625.
What differentiated Nestorians from the Chalcedonian Churches, and why they were basically booted out of the councils was their understanding of Christology. Unlike Egypt which emphasized the Divine Nature of Christ, almost the exlcusion of his human side, The Nestorians went in the other direction - emphasizing his humanity to the point where they claimed that Jesus had two completely separate natures - as if he were what we today would call schizophrenic. In some later incarnations, the idea was even put forth that Jesus and the Christ were two distinct things, with the Christ occasionally appearing with in Jesus.
In the end, however, Islam first co-opted, then supplanted Nestorianism pretty much everywhere it went.
There were probably some groups in eastern Asia, like India and China. If I recall correctly, there are the Jesus Sutras which were finished around 640 CE. There are also those like Bardaisan(sp?) who lived around the time of one of the councils but were farther east.
Yes: the Apostle Thomas evangelized a significant part of the globe, including north and south America and Asia. A summary, drawn principally from the writings of Cornelius a Lapide and Dr. Warren Carroll -- who cite that the Fathers and Doctors of the Church teach that the Gospel was preached literally to the whole world before the death of the last Apostle -- is recounted in this sermon/homily on St. Thomas the Apostle. I don't think many/any of these evangelized people attended any of the first four Councils.
UPDATE: For those down-voting this answer solely because I'm citing the Apostle Thomas' evangelization of north and south America, here's a link for you: http://is.gd/yAnaJg -- and notice that the Apostle Bartholomew is recorded to have been with him. More links to come as I find them. If I have to scan printed texts and link to those I will.