"To whom shall we go?" - John 6:68
Often when a specific prayer goes unanswered we become angry with God or disillusioned with prayer. After all, we didn't get what we wanted! But anger is not a viable long-term position. Not because of what it will do to your soul (and it will destroy your soul...), but because if a God exists then this being by definition will have reasons for not answering your prayer that you know nothing about. Indeed, he may have reasons for not answering your prayer that you couldn't or wouldn't understand if you did know them.
Ultimately, you must pick atheism or submission because the middle ground is unstable. So for the "already convinced" who have decided that they have not enough faith to be an atheist, the solution can't be "just don't be angry." Try telling that to an angry person! Instead you must pour out your anger, indignation, or disillusionment to God. Eventually in this process you'll discover the one thing of greater value than that for which you have prayed: God himself. He is the one treasure worth seeking in prayer above any other thing.