I have had a serious prayer life for 26 years. Every week, every year, for 26 years, I take several hours to go in solitude to pray. Have I ever felt that God has not heard my prayers? It strikes me that I have rarely felt like crying out "God, why are you not listening to me!"
I have met people who tell me "I prayed, but God did not answer my prayer." Or, more strongly, I have heard the atheistic statement "Prayer does nothing, accomplishes nothing. Prayer does not "work"." I understand what is being said. I believe I can feel compassion towards such things. But, honestly, they do not move me very much. I think I never feel or have felt that prayer accomplishes nothing. If I did seriously entertain such a possibility I feel certain that I would stop praying.
I heard an "ex-Christian-become-atheist" once say, "I was in deep trouble. I prayed to God. I received no answer." The end result for this person was that they renounced Christianity and now spend a portion of their life trying to make their atheistic point against Christianity. (Which, I confess, still strikes me as an odd thing to do. At least I feel that if I became an atheist I would not waste my time arguing against Christians and theists, since I would find such activity, like all activity, ultimately meaningless. Surely there would be far better things to do in life? Unless I had some unconscious psychological need to defend my new noetic framework... )
Only a person who has a serious, committed, long-term prayer life has an authentic right to declare that "God does not answer prayers." The "Christian" who throws up occasional prayers and "is too busy to pray" is like a guitarist who never practices yet declares "Practicing does no good." How would he know?
Personally, for 26 years I have been and am involved in a great prayer experiment. I rarely think that God is not responding to my prayers, and often think that He is responding. Only someone who really, actually prays over a lifetime can discover such a thing. It reminds me of C.S. Lewis when he writes of the logic of personal relationships which creates "obstinacy in belief."