There are many verses even in the New testament, that tell us to purify our hearts. So If our hearts are totally clean through the new birth it would make no sense to purify them any further.
"Draw nigh to God, and he will draw nigh to you. Cleanse your hands, ye sinners; and purify your hearts, ye double minded. " - James 4:8
James is writing his epistles to born again christians. There are more verse, for example Hebr 10:22, who is telling us to sprinkle our hearts clean, 1. Ti1:5 tells us that the END OF THE COMMANDMENT is love out of a PURE HEART, etc
And then you have the other side, like Act 15:9, where it says, that it is GOD who PURIFIES THE HEART. There are few preachers who preach both sides of this truths - most of them tend to lean to one side.
(Actually the theology concerning the "flesh" is the same - on the one side is our flesh ARE ALREADY CRUCIFIED WITH CHRIST (past) and then there are commands to mortify flesh)
This is the same as in the Old Testament: There it says in many verses (you can find them with eSword) that God will "give us a new heart" or "a clean heart", that is the part that only God can do. And then it says, that we have to "circumcise the foreskin of our heart" - That is our part. God will never do our part and it is impossible for man himself to do gods part. Only God can give a new heart. But we have to do the cirumcising, that is the denying of self, the forgiving, the humbling, the repenting.
So as we humble ourselfs and clean our hearts, God will give us more and more a pure heart.
Brethren, don't be satisfied with mere theological understanding of this topic. Proverb 4:23 speaks of the importance of the heart, and Jesus mentioned that OUT OF THE HEART PROCEEDES every thing and that the MOUTH SPEAKS out auf THE ABUNDANCE OF THE HEART.
If our mouth speaks evil things, than don't come with theology and say "but I have a clean heart". You have not. Humble yourself, repent and God will make your heart cleaner than ever before.