One composer who felt this atonal ambivalence with special acuteness was the American George Perle (1915-2009). Enraptured with the wide open possibilities of Viennese atonality from an early age, Perle nevertheless felt Schoenberg left behind an unfinished project. Perle believed that Schoenberg's 12-Tone Method (also known as serialism) was a partial and unsatisfactory way for dealing with these new resources. As prominent a music theorist as he was a composer, Perle set out to discover a more musical basis for picking this note over that. In a series of influential books and articles, he ended up formulating what he called "Twelve Tone Tonality," a system that allowed for many of the gestures of traditional music -- cadences, "mode" and "keys", and most importantly a real sense of directed chord progression -- without any of the trappings of tonal pitch selection or syntax. Perle wrote the majority of his own music according to this system. But rather than existing simply to validate his music theoretically, Perle's special brand of "Tonality" was deeply committed to accessibility.
Someone suspicious of atonality and its reputation of ugliness and difficulty would do well to give Perle's Sinfonietta II a listen. He composed this in the fall of 1990 during his three year tenure as the San Francisco Symphony's composer-in-residence. It was performed in February of the next year under the baton of Herbert Blomstedt. Newspaper reviews were quite lavish with their praise, and especially quick to emphasize how enjoyable, or in the words of Sacramento Bee reviewer William Glackin, how "happy" a work it was. The San Francisco Examiner called the Sinfonietta "disarmingly communicative" (expecting something else?), while the San Jose Mercury News basically dubbed Perle the spiritual successor of France's "Les Six": "Nothing is trite; nothing is a rerun; nothing panders to popular taste."
Like the duration (a fleeting 16 minutes), the orchestra of Perle's Sinfonietta II is quite small, save for the percussion section, which keeps its four performers busy with thirteen colorful instruments in strategic reserve. The outer movements, labelled Scherzo I and Scherzo II, are siblings in terms of structure and ethos, both playful romps with a clear ABA form. Orchestration is usually tidy and open, the opposite of Berg and Schoenberg's dense-thicket approach, closer in spirit to Webern's crystal lattices. But that's basically where the Webern similarities end, because Perle projects a sense of fun and extroversion completely alien to that more rarefied style. Melodies and motifs crop up repeatedly, meaning that Perle keeps the listener from worrying about where they are in the piece. The attention span of these ideas is quite short, leading to a manic, scurrying quality that persists throughout -- at several times, especially in the second Scherzo, there is a strong suggestion of cartoon music!
The first Scherzo sets off with a handful of tricks: first a climb up an arpeggiated diminished seventh (I mean, err, presentation of the interval-3 cycle mode-1...), then a harmonized flute phrase, a pair of cello interrogations followed by more arpeggios, and a set of strident but not-too-dissonant string chords:
The harmonic materials, the play of rhythms, and the quickly alternating orchestral colors are characteristic of the whole movement. The B-section "trio" introduces a nervous triplet pattern for strings, over which the flute and piccolo rush in a panic, soothed and egged on at intervals by vibraphone and brass whole-tone arpeggios.
To me, the second Scherzo is the funnier of the two. There is right off the bat a strong contrast between annoyed insistence (the strings' upward bounding, dissonant theme) and mellow sheepishness (the wind and pizzicato answers). After each urgent string statement, a corresponding gesture of deflation seems to follow. Particularly striking are the unvarnished clarinet major thirds and the short passages of honest-to-goodness jazz scoring for brass and percussion -- what better way to puncture an insecure atonal phrase's sense of angsty importance?!:
I have no doubt this work's pitch design is rigorously organized, but the impression is the exact opposite of what you'd expect coming from a theorist of 12-tone quasi-mathematical structure: this is academic atonality, thoroughly defanged.
The heart of this short work is the second movement, "Chorales and Diversions." Hearing it on Pandora (on my Nørgård channel, no less) first made me interested in Perle's Sinfonietta, and it is the movement I find myself returning to the most. A rondo in form, the movement sandwiches some eerie passages of rather Bergian sound between statements of a lyrical chorale melody, given almost exclusively to a solo bucket-muted bass trombone. The theme respires in and out like a Bach chorale glimpsed in the dusk, and is harmonized homophonically with some truly lush lower string writing. Here is the entire first section of this nocturnal movement:
George Perle may be the only composer of resolutely "atonal" music for whom I've noticed the label "conservative" repeatedly applied. Sometimes this is in reference to his choice of musical forms. But elsewhere it seems "conservative" is the only word we have for composers who attempt to eschew abstraction for its own sake, and instead choose to ground their music in a consistent and accessible system like traditional tonality. Ultimately, it's as unsatisfyingly simplistic a label as would be "progressive." Better to take Perle at his own word:
"There's this mystique that there's an elite of specialists for whom contemporary music is written. I don't write up or down to anyone. I'm just doing what composers have always done. Some people have written about me as though I were a composer of inaccessible music. But my experience has been that people who listen to my music are amazed by how accessible they find it." 
1: The foundations of Perle's theory are two interlocking organizational parameters. The first is the inversional array, which relates notes of the chromatic scale to stable pitch axes (often about a tritone) - this hinge is his equivalent to "key." The second is the interval cycle, which selects a number of pitches based on some symmetrical partition of the scale, such as minor thirds or major seconds - this furnishes the sense of "mode." Both are prominent aspects of the styles of not just Berg and co., but of Bartok, Debussy, and even composers further back into the 19th Century. In compositional practice (and Perle's analyses) the interactions of these two factors can become exceedingly complex, but the important point is that they help produce points of reference and stability that keep the listener from ever becoming too lost.
2: Reviews excerpted at GeorgePerle.net (not surprising that they are so positive!)
3: Quoted in For the Love of Music: Invitations to Listening (Steinberg and Rothe, 147)