Here is my quick take on Lévi-Strauss's contribution to Media Ecology:
His approach to the interpretation of so-called "primitive" cultures revealed the complex patterns of thought that went into the development of systems of myth and kinship. His notion that "primitive" intellectual activities were equal to our "modern" systems of knowledge, just applied to differing objects, put the entire body of anthropological writings, going back to James Fraser's Golden Bough into a new perspective. Lévi-Strauss's work, building on the linguistic structuralism of Ferdinand de Saussure and the evolutionary approach to cultural studies of Franz Boas, revealed the biases in the use of terms such as "primitive" and "modern" (even though Lévi-Strauss himself used these terms), and paved the way for Walter Ong's distinction between orality, literacy and secondary-orality as more appropriate explanations for differing cultures.
Lévi-Strauss discovered and demonstrated connections between seemingly disparate mythic stories, and offered explanations for seemingly random elements of those stories. His methodology can be used as model for ways to interpret the products of contemporary culture, which, while seeming to be unrelated, actually constitute a system (or systems) of symbolic meanings.
I find the tools Lévi-Strauss provides useful in a number of ways. I also think that his notion of "things that are good to think with" as powerful as Postman's question regarding a new technology: "What problem does it provide a solution to?"
I like Lévi-Strauss's idea that a myth is not a "false" story or idle tale, but rather a dynamic technique which members of a culture use to address cultural discrepancies. To me, this is a compelling explanation for why myths persist in a culture. It may also explain how, if we know where to look, we can identify the mythic systems of our own culture that provide us with a coherent world view in the face of constant change and turmoil.
Myths cannot be considered singly. One must absorb the entire canon of a culture to understand all the connections and interactions. That may be why we can't analyze Greek mythology properly. Too little of it has come down to us, and the versions we read have undergone so many revisions that they may not truly represent the originals.
I know that structuralism has been somewhat in eclipse in academic circles lately; that Lévi-Strauss has been accused of a binary focus that, being Hegelian in origin, cannot apply to current thinking about media. I think that to write off Lévi-Strauss's methodology as a thesis/antithesis/synthesis intellectual game misses the subtlety of his analysis. A closer reading of all 2200+ pages of his Mythologiques shows that, while he may begin an analysis by identifying polar opposites, this is only a starting point. The analysis of a mythic system must account for far more that just a pair of opposites. In the course of his analysis of the myths of the Tupi Indians, Lévi-Strauss moves spiral-like through multiple mythic variations and multiple opposing pairs and by proceeding A to B and B to C, etc., demonstrates internal consistencies within the mythic system that aren't immediately apparent to an outside observer. In other words, Lévi-Strauss provides a useful tool for analysis regardless of whether you wish to extrapolate the function of the method to the deeper structures of the human mind or not.
I also find Lévi-Strauss's methodology completely compatible with McLuhan's Laws of the Media. Where McLuhan, via the Tetrad asks us to consider what a technology or medium enhances, obsolesces, retrieves and reverses into, Lévi-Strauss will start with a pair of opposites "A" and "B", but in the course of his analysis will present examples of what he calls " A' " (A prime) and " B' " (B prime) as recursive iterations of the original pair. Perhaps someone will someday conduct a Lévi-Straussian analysis of McLuhan's system of myths.
Lévi-Strauss is also compatible with Ong's notions of primary orality. Ong discusses how different human thought processes must be without text. Lévi-Strauss gives example after example of exactly how these thought processes work. I don't recall Lévi-Strauss discussing the poetry of the Tupi Indians as a means of perpetuating the culture, but he does demonstrate how interconnected myths can act as intellectual place holders for a non-literate population to help them consider complex systems of thought.
Our current system of myths is not only presented verbally, but also via images, in print, and even in interplay of biases amongst all of our competing media. Since we may be relearning this manner of thinking as we move deeper into secondary orality, Lévi-Strauss provides us with a map of where we may be headed.