The Book of Psalms is a collection of the prayers of the Hebrew people. As such, it would have been Jesus' principle prayer book. It is natural to assume that he was taught these prayers as a youth, and that they continued to play an important rôle in his adult spirituality. This assumption is proven to be fact on the cross, when Jesus turns to one of the psalms to express the inexpressible:

My God, my God, why have you forsaken me? Why are you so far from helping me, and from the words of my groaning? 
—Psalm 22:1

Growing up, I used to hear it said that, since Jesus became our sin (see 2 Cor. 5:21), his quotation of this verse needs to be taken literally—that indeed God forsook him completely because God could not countenance sin. I have not adopted this view as an adult. If I did, I would be endorsing two things I do not believe:
  1. That the psalter is not for everyone—that this psalm of anguish in particular can only be quoted by those whom God ignores.
  2. That God would reject anyone.
Instead of expressing a bizarre theory of God refusing to look on a suffering Jesus, the opening verse of the 22nd Psalm is the most reassuring evidence we have that the psalms are real prayers, that the emotions they express can be laid before God, and that (and this we find out as the psalm progresses) what seems to be the end is not the end, as long as God is involved.

After all is said and done, I think it's most important to remember that God is not limited by my emotions. God is greater than anything I can feel, and just because I feel like my prayers are falling on deaf ears does not mean that God cannot hear me.

Eternal God, your tortured Son felt abandoned, and cried out in anguish from the cross, yet you delivered him. He overcame the bonds of death and rose in triumph from the grave. Do not hide your face from those who cry out to you. Feed the hungry, strengthen the weak, and break the chains of the oppressed, that your people may rejoice in your saving deeds. This I pray in the Name of him who taught me to pray: Our Father...
—(Roman Catholic) Liturgy of the Hours