Jesus tells us in Matthew 21:22, "And whatever things you ask in prayer, believing you will receive."
When we pray, God works. Just because we don't know much about prayer and how it works that is no reason for us to stop praying. We don't know much about the soil we stand on, but that doesn't prevent us from walking on it.
One thing we do know is that the Holy Spirit puts fresh ideas into our minds when we pray. It's obvious that Satan fills our minds with bad (negative, confusing, distracting, sinful) thoughts. So then, isn't it just as obvious that God can give us good (positive, hopeful, productive, others-centered) thoughts?
We often wonder why God leads us through such difficult times. But God knows that when He pieces our trials all together in just the right way, things work out for our good. We just have to trust in Him and, eventually, our problems will turn into something satisfying for us. God loves us or He wouldn't keep sending us flowers every spring.
We all believe that God can, in answer to prayer, heal the body, prevent tragedies, and send rain. Sometimes He sees that we desire the right thing but we are going about it in the wrong way. The question is not, "How does God answer our prayers?" The question is, "Do we really pray?" Do we give God the opportunity to put His will into our hearts?