The pieces I really can't get into are the atonal dodecocacolaphonical stuff or whatever it's called; Schönberg, (maybe Berg, but I haven't heard any pieces by him) and Webern. Schönberg's Pieces for piano I find rather annoying and detestable, but my list with worst piano compositions ever tops with Webern's Varations op. 27. Even Richter played them, but they are so awful to listen to!
Must remember that one.
Can I suggest dodecacophonical as an alternative ?
Yes I agree, that system, as a means in itself, has proven to be sort of a dead-end. However its influence on other composers has been lasting, and it was perhaps a necessary step in the evolution of music.
Neither do I really like Barber, Cage, Copland and those American composers.
That's a bit of a sweeping statement. Many 'modern' American composers actually write very accessible and tuneful music. Barber can be a bit acidic, even atonal, but also gorgeously romantic (listen to his violin concerto, the symphonies, and the School for Scandal overture - not to mention that Adagio for Strings which is all but played into the ground). His piano concerto I find hard going though. The Sonata is a masterpiece of contemporary piano literature.
Copland can be really abstract and even atonal at times. But he has lots of immensely appealing works - Rodeo, El Salon Mexico, Danzon Cubano, the Third Symphony, Appalachian Spring - in short, his "Americana" works that everybody likes. Notsure about Cage - I think he's more of a prankster than a serious and lasting composer.
And among "those American composers" you find many similar names, composers who write gorgeous music in traditional mould, like Creston, Hanson, Diamond, Piston, and many more.
I prefer orchestral or chamber music in the contemporary repetoire. A piano concerto sounds much better than the pieces for piano solo written today.
Not sure where you get that idea from. You could check Stephen Hough's album "New York Composes" with works by Copland, Ben Weber, Corigliano, and Tsontakis. Especially the latter two are very accessible and great to listen two. But the other two are worth the effort too.