Elijah's expectation of rain after the great fire-miracle at Mount Carmel (1st kings 18), is a good example of having an expectant attitude in prayer. But the story also warns us against being presumptuous in prayer.
Elijah was confident that the Lord was sending rain because the word of God told him so -- in this case, a personal, prophetic word. God has given us His written word, the Bible, as our authoritative guide for prayer. Elijah could be confident because he understood God's word correctly. This is a timeless rule for us.
There are many people who claim or declare things in prayer that are not taught in God's Word. For example, God's blessings on Jabez (1st Chronicles 4) are not God's promises to us. The story illustrates God's goodness, and prayer's efficacy, but it doesn't prove that God wants to increase our real estate holdings, which was one of the blessings Jabez asked for.
In the same way, Psalm 103:3 does not say that God's will is to heal all Christians' diseases at all times. We know that is not so because we study the entirety of the Bible, and pull all the data together and integrate it. Psalm 103:3 tells us what God's general will is, and the rest of Scripture tells us there are at least eight exceptions to the general promise. It's wrong to beliefs about healing on one or two verses.
In the same way, for a Christian to be legitimately expectant of God taking action, he must be interpreting a promise correctly.
Second, expectancy isn't based on how fast God answers, or what methods He uses. When Elijah prayed at the altar, God's miraculous fire fell down immediately, consumed the offering, and licked up the water. But when Elijah prayed for rain on top of Mount Carmel, his servant had to check seven times.
Finally, on the seventh try, the servant saw a little cloud forming out over the Mediterranean Sea. Two different prayers led to two different modes of answer -- different in speed, and different in manifestation (the first instant and outwardly miraculous, the second slightly delayed and outwardly natural). Don't tie your faith to your own concepts of how God will answer.
Last, Elijah kept praying even while the answer was coming. He went to the mountaintop, knelt down, and prayed even though he had full confidence the answer was already on its way. It isn't the sin of unbelief to keep on praying. Christ taught against vain repetition in prayer (Matt. 6:7), as if incessant jabbering in God's ear moves Him to action. But Christ never said to not pray.
Pray with expectancy, but make sure you aren't claiming something God never promised. Make sure you aren't basing your beliefs on just one verse here or there, because that's almost always the path to error. And don't anchor your faith to how you imagine God will answer. God can do exceeding-abundantly above all that we can ask or even think.