Do our prayers do anything?

I was amazed at a recent discussion about prayer, in our Bible study. Here's a boiled-down version of the conversation:

J: If I pray secretly for 'Fred', can those prayers help him?
Z: Yes.

J: So, if I pray for Fred, God blesses him through my prayers?
Z: Sure.

J: Then, if I refuse to pray for Fred, he doesn't get those blessings, right?
Z: Wrong. God can still bless him without Your prayers.

J: Of course God can still bless him. God can bless Fred in lots of ways, through different instruments.
Z: Exactly.

J: But I thought You just said that Fred actually does benefit as a result of my prayers.
Z: He does. But God can bless Fred without waiting for Your prayers.

J: Either our prayers bring about blessings from God or they don't. Scripture says that our prayers are effective. They actually bring about good. If I refuse to pray, I'm refusing to let God bless Fred through my prayers. I'm preventing God from blessing Fred through one of the instruments that God specifically wants to use for Fred's blessing.
Z: You can't block God.

J: Yes I can. And I do, whenever I sin. Sin prevents God's will from being carried out in my life, and this can and does hurt others too. If I refuse to pray for Fred, I'm blocking one channel of God's grace into Fred's life. I'm not preventing God from blessing Fred in other ways, but I am preventing Him from blessing Fred in one important way: through me.
Z: But God doesn't need Your prayers. He can bless even if nobody prays.

J: You're going in circles. You can't have it both ways, Z. Scripture says prayer is powerful and effective. Consequently, a lack of prayer causes a lack of some good and truly important effect that God desires to accomplish. If You say that my refusal to pray has no affect on Fred, then You can't claim that my prayers can do him any good. If You take that position, Your view of prayer is at odds with Scripture.