My own personal opinion on this matter;
Duas in my experience are accepted, always! It just depends on the dua tho, I've made the idle oh I wish type of dua which is half hearted and I don't really expect or even much want it to happen.
Then there's the dua I've really honestly whole heartedly made, and these have come true almost immediately, however mostly in ways I completely did not expect.
There have been the duas I made whole heartedly and fervently, which were accepted after a little time (altho to me at the time it felt like a very long time).
Then there's the duas (I make loads of duas OK), I've made and really really felt I needed them to be answered and after making the dua, instead of an immediate material answer, the pressing fear/terror/sorrow which had inspired the dua is suddenly lifted and my heart feels lighter. Not to say I then stop making the dua, but I'm comforted with an unshakeable certainity that that which I have asked for will be OK......
Personally I think all duas are answered, we are too impatient, and sometimes we are so set on the way we want our answers, we fail to see the reply to our duas because the reply was not in the form we expected.
And sometimes I make dua just for the sake of feeling close to Allah, because sometimes Salat is completed just too fast and I want to remain in the presence and the closeness of Allah swt for a little teeny bit longer......