Jesus says ask and it will be given to you.
He does not say when. He does not say how.
So yes, of course we believe prayer works, granted that we understand the nature of God, and we are patient and try to see things outside of time, from God's perspective.
If we really understand who God is - that he is love - then we understand that in God's very nature, he cannot grant anything that would contradict his nature. For example, if we request something to satisfy our personal, selfish desires, then how can a good God possibly grant such a request? It would contradict his very nature.
But if we asked for something like peace in the world, then of course God can do that. And we must believe it will be granted. It is us desiring for something good, for restoration, for beauty.
But there is a process.
We have to trust the process.
If we ask for a sick child to be healed from a disease, of course this is a good thing. But sometimes all we can see is things from our perspective. You see, from God's perspective, that child has already been healed and raised from the dead and is living life eternally. But from our perspective, bound by time, sometimes we don't see the big picture. That’s why we always say “Your will be done.” That’s how Christ, our perfect example, prayed.
In each Divine Liturgy, we ask for salvation, peace, forgiveness and healing. You have to imagine that when we are in church, we are praying and groaning with creation from the beginning of time. The Divine Liturgy is meant to transcend time, be eternal. So when you really think about it, we have already been granted these things we constantly ask for. God has already restored everything and everyone through the incarnation and resurrection. He has restored our lives and raised up the entire creation. All our petitions have been granted. We ask him to save us; he will and he has. We ask him for deliverance; he will and he has. We ask him to be forgiven; he will and he has.
And THIS is the point of prayer. That we are reminded of all this. We ask all these things, and he says, yes my child, patience; it has, it is, and it will be done.