This is a favorite topic for a lot of people, and there is regular ranting & raving on issues somehow related to it, so perhaps some remarks are in order. As most musicians know, tonality is a complicated phenomenon. That's not to say the "you know what I mean" crowd is totally off base, but it does require some thought to determine what they do indeed mean. Melody accompanied by harmony is nearly unique to post-1600 Western music, although there might be some other musical systems (notably Southeast Asia) which can possibly be construed that way. Once we get into modulation and key systems, in short functional harmony, there is nothing else like it aside from some derivative fusion efforts around the world. So what do the people who rail against "atonal" music actually mean? Extensive experience with early & world music perhaps gives me a different perspective on what is otherwise an overly discussed topic.
Attempts to precisely define tonality account for a great deal of verbiage, so there would be no sense attempting to come to a conclusion in the space of this column. I prefer a fuzzier approach, where we know that Haydn & Mozart are tonal, and then can relate similar approaches to varying degrees. Anyway, the point is that Medieval & early Renaissance music are not "tonal" in any technical sense of the word. What then are the people who complain about "atonality" talking about? Mostly I think they use "tonal" as more or less equivalent to "the organization of music I like", without any technical connotations. If questioned about Renaissance music, responses are typically of the sort "modal then, same thing." Chant is generally modal, certainly, and while we might be tempted to lump this together with tonal, the remarks are never directed toward monophonic music. Assigning a mode to polyphony is essentially inelegant, and often impossible. It already stretches a monophonic classification system beyond the breaking point. If we look at Gombert, who has been mentioned here before, the idea that his music is tonal would invalidate any vertical restrictions on tonality, since every interval appears, frequently unresolved. In Medieval music, the cadential motion is quite different, lacking structural thirds. For Bach, as is so often pointed out, there is a wealth of dissonance. Even if we start speaking of "percentage of dissonance," as some like to do, that gets us nowhere with Gombert et al., not to mention that "dissonance" is a term with a technical meaning only within the context of a musical system.
The sheer multiplicity of vertical elements leaves us with the linear aspects of tonality, and that is perhaps a good place to turn. A "hummable melody" (why humming?) is often given as another requirement alongside demands for tonality. I do think this aspect is probably more consistent than the vertical one. Given the frequency of such utterances, I am disinclined to construe them as meaningless, and indeed such a stance is typical of my approach. Since it is a general feature of diatribes against "atonality" that your average person on the street would never sing such a thing naturally, it is integral to the thesis that early & world music be "tonal" in some vague sense, so we must look there for clues as to what is meant. Early music is hummable at the linear level, and indeed it is "classical tonality" which more often includes a part without tunes. The honest classical musician will readily admit to finding early music more difficult, due to its lack of harmonic center. Things happen faster. Of course, that fits well with the idea of harmonic accompaniment as a simplification for the development of multiple voices. I am not really sure how this greater linear complexity affects the relationship of early music to "tonality" in the vague sense, but that is a topic which merits future discussion. World music likewise has hummable melodies, alongside purely rhythmic parts. The rhythmic parts appear to present no problems for most listeners, and indeed the variety there has apparently been the major force behind world music's current popularity. The melodies can be thornier in their relationship to vague "tonality."
Finding examples of art music from other cultures dissonant to the Western ear has never been a challenge. Simple differences in tuning are enough to frighten many people, although the various systems used by the early music movement have softened that aspect somewhat. Turning to more specific examples, the Japanese court orchestra music known as Gagaku is extremely dissonant, dwelling on microtonal clashes and even employing them cadentially. This is a case of both linear & vertical dissonance. For melodic examples, Indian music is always a place to turn, as it has an uncommon linear richness. Indeed, in Carnatic music, there are melody-patterns (i.e. ragas, which are somewhat different from the Western idea of "mode") which are specifically designated as dissonant. These are the "vivadi ragas," containing intervals smaller than a semitone. Several such ragas have long histories and are frequently performed in concerts. Indeed, as this is primarily a vocal repertory, these melodies are very hummable. In fact, I find quite a bit of Schoenberg's music hummable as well. The key here, in my view, is that these melodies are actually conceived as melodies and are given a linear continuity. It is not the set of pitches so much as the way they are connected.
Without implying any value judgements, this leads to the basic conclusion that, aside from the more basic "music I don't like," objections of this type are related more specifically to linear dissociation. Certainly Boulez or some of the other modernist composers do not offer melodic continuity, and this seems a definite factor in their reception. The style also yields a different perspective on vertical relationships, because the individual lines do not have a clear direction to imply a normal resolution of "dissonance" or even a meaning for the term within the basic scheme. Frankly, I have trouble hearing any combination of ordinary chromatic notes as physically dissonant, and that includes clusters. The difficulty of the music comes from developing pitch relationships over longer spans of time, such that they cannot be expressed in more compact terms (i.e as a melody). Perhaps a more interesting observation with respect to this article is that pointillistic styles appeared before, in the early 15th century. This is most marked in the mass movement settings of Pycard and other English composers, but is also found to some degree in some Italian settings of the era. Oddly, some of the most extreme examples of pointillism come from liturgical settings, perhaps finding inspiration in the rather disjointed texts of the Credo & Gloria.
As it comes time for a conclusion, there isn't one to be found. Perhaps some other thoughts will be forthcoming on this issue, but for now I think it only worth noting that music appreciation is obviously learned, a fact which helps to explain the ready reception "atonal music" finds in films and other context-laden situations.