After attending and performing at most of this week's New Music Festival in Winnipeg, I was forced to conclude that most people don't understand what "new music" is. When most people think of "new music" in the Western art world (read: Classical music), they think of really weird, almost unlistenable sounds.
Here is a quick primer to understand what makes "new music" tick. You might even find something to use in a non-classical music style as well!
New Music is based on this principle: All sound is music. This principle came out of composers trying to take music in new directions after the first World War. This led first to new systems, such as Schoenberg's twelve-tone rows, and later, the serialism of Boulez and Stockhausen, and also to new conceptions of what music should sound like. A turning point in this stream is Varese's Ionization, which was written for a percussion ensemble.
After Varese, composers took the sound of music even further - people began writing music that included extended playing techniques, such as clicking the keys on a flute, or tapping a violin body, or even hitting the strings inside of a piano with a mallet! This led to a greatly expanded sound palette that composers could draw from, and an increased understanding of how a sound or texture could be a melody (rather than a series of notes).
I could go into more detail, but for the sake of keeping your interest, I'll stop here and let you read up on things for yourself if you are interested - Wiki John Cage for a good read.
So, to understand listening to "new music", first think this: What do I hear? Don't judge immediately. Try to identify the sounds that you are hearing. Also listen for musical ideas that keep coming back. Next, think "What does this sound like?". Does this sound mimic anything I have heard before? What kind of emotions or feelings does this sound evoke in me? After you have thought about those things, then you can judge for yourself whether you appreciated the piece.
Now go out and listen to some new music!