In Acts 10 the Bible relates the story of Cornelius, the Roman centurion, a Gentile who was visited by an angel and commanded to send for Peter, who would give him instructions. At God's direction, Peter agrees to go to Cornelius despite the fact that, as a Jew, he generally avoids the company of non-Jews. After finally hearing Cornelius' story, Peter is duly amazed and exclaims: Of a truth I perceive that God is no respecter of persons (verse 34).
This account has several interesting aspects. If one wonders why Peter would have initially thought God was a respecter of persons, you will find the answer in the Old Testament. Strict segregation from non-Jews was an issue throughout the Hebrew scriptures, so that idea of exclusiveness and separation was an integral part of Peter's Jewish heritage and religion. It, therefore, took a special divine visitation to prepare Peter for the change of mind and heart which allowed him to overcome his Gentile prejudice and go to Cornelius.
The typical lesson drawn from this Bible story is that after Christ, Peter was operating under new rules, such that Gentile believers too could enjoy God's favor just like Jewish believers. Thus, Jesus supposedly allowed God to be no respecter of persons after he died and rose again. From Peter's initial reaction we would logically conclude that he did not understand that change as having occurred post Christ, until he had this special revelation.
One might also ask how it is that Cornelius would be approached directly by God based on his being a good man, though a non-believer, and then evidently anointed by God prior to completing the typical church mandated salvation process. The general theology of the church today would not allow such to be recognized and sanctioned.
If in response, we conclude that this is a special case, outside what could happen now, we draw a red question mark over Peter's quote in verse 34. Instead of the incident proving that God is not a respecter of persons in relation to Jew and Gentile, it demonstrates that for Cornelius special case He chose to be a respecter of persons to make His point. If this conclusion be true, Peter's declaration is strangely amiss.
Orthodox Christianity really doesn't allow God to be no respecter of persons. First of all He supposedly discriminates against non-believers. Then, God judges the works of every man, even believers, who enjoy differing degrees of reward in the hereafter. Peter may have been astounded by a new revelation of God in Christ, one that eliminated the prejudice, discrimination, and rejection of the past, but the church doesn't let that astonishment get in the way of its theology.