That last post from Lousyreeds cuts to the heart of this issue very nicely, though might be a bit hard on Robm 321.
At the outset, let me say that I am not a musician and my last music class was in the 7th grade. My parents weren't interested in serious music, so I developed an interest almost by chance. My first purchases were a set of Beethoven symphonies, the Bach St. John Passion, and the Mahler 9th, (for a total of $20). Pretty quickly I discovered Stravinsky, which really primed the "modern" pump for me, but it took a long time before I could really enjoy Mahler. Sometime later I started attending live concerts. Over the last 35 years, I have developed a great love for a wide variety of music from Heinrich Schutz to John Adams. I spend as much time listening to Shostakovich as I do Bach, and love them both equally --- but--- I bet I've never listened to them both in the same sitting.
I heard the world premier of Lohrmann's Symphonisches Stuck this year. In the preconcert lecture, the interviewer asked Lohrmann what the audience should listen for. He answered, "Don't listen for anything, you either feel the music or you don't." I also heard Berg's Three Pieces for Orchestra this year. The conductor offered this advice to the audience. "I programmed this work because every musician should play this music once in his career. All of you should hear this music. This is difficult music. It is like Goethe's Faust. One does not really like Faust, but one should be aware of it."
These are two very different points of view. Most listeners tend to approach music from the former point of view. We are tired, stressed, harried, and harassed, more in need of a Haydn Missa Brevis than a difficult and disturbing mystery. In making this choice, we gain something, but also loose an opportunity. Much of the music of the last 100 years is worthwhile, important, and deserves to be played, heard, and recorded.
I find the body of serialism uniformly difficult. For me, these works require a live performance where there is more to the experience than following the tune in one's listening room. Even then, it takes wisdom in making programming choices, hard work from the musicians, and careful explanation in pre-concert lectures and program notes. With enough similar exposure, I might begin to understand this music, and be the better for the effort it takes. However, as I look around the concert hall, I notice that the response is tepid, at best. I also notice that most of the audience probably won't be around 20 years from now, and that worries me very much. Alas, free market forces are everywhere, even where they probably shouldn't be.
So, in answer to the call for accessible modern music, let me toss out a few suggestions. Shostakovich quartets, especially 6, 8, and 10. Prokofiev Symphony #5. Bartok, Divermento for String Orchestra. John Adams, almost anything. The Copland piano concerto--fun music!