In John 3, Jesus is discussing the new birth with Nicodemus, and is "amazed" (my quotes) at the fact that Nicodemus did not understand this- He says in verse 10,"you are Israel's teacher and you do not understand these things." Where in the Old Testament is the new birth described in such detail that Jesus could have assumed Nicodemus should know about it?

The idea that Israel would have to have a new heart before they could enter the kingdom should not have been a new idea to a real student of the Old Testament scriptures. Carefully read the following passage:

Ezekiel 36:23-35 [23] And I will sanctify my great name, which was profaned among the heathen, which ye have profaned in the midst of them; and the heathen shall know that I am the LORD, saith the Lord GOD, when I shall be sanctified in you before their eyes. [24] For I will take you from among the heathen, and gather you out of all countries, and will bring you into your own land. [25] Then will I sprinkle clean water upon you, and ye shall be clean: from all your filthiness, and from all your idols, will I cleanse you. [26] A new heart also will I give you, and a new spirit will I put within you: and I will take away the stony heart out of your flesh, and I will give you an heart of flesh. [27] And I will put my spirit within you, and cause you to walk in my statutes, and ye shall keep my judgments, and do them. [28] And ye shall dwell in the land that I gave to your fathers; and ye shall be my people, and I will be your God. [29] I will also save you from all your uncleannesses: and I will call for the corn, and will increase it, and lay no famine upon you. [30] And I will multiply the fruit of the tree, and the increase of the field, that ye shall receive no more reproach of famine among the heathen. [31] Then shall ye remember your own evil ways, and your doings that were not good, and shall lothe yourselves in your own sight for your iniquities and for your abominations. [32] Not for your sakes do I this, saith the Lord GOD, be it known unto you: be ashamed and confounded for your own ways, O house of Israel. [33] Thus saith the Lord GOD; In the day that I shall have cleansed you from all your iniquities I will also cause you to dwell in the cities, and the wastes shall be builded. [34] And the desolate land shall be tilled, whereas it lay desolate in the sight of all that passed by. [35] And they shall say, This land that was desolate is become like the garden of Eden; and the waste and desolate and ruined cities are become fenced, and are inhabited.
See All... And I will sanctify my great name, which was profaned among the heathen, which ye have profaned in the midst of them; and the heathen shall know that I am the LORD, saith the Lord GOD, when I shall be sanctified in you before their eyes. For I will take you from among the heathen, and gather you out of all countries, and will bring you into your own land. Then will I sprinkle clean water upon you, and ye shall be clean: from all your filthiness, and from all your idols, will I cleanse you. A new heart also will I give you, and a new spirit will I put within you: and I will take away the stony heart out of your flesh, and I will give you an heart of flesh. And I will put my spirit within you, and cause you to walk in my statutes, and ye shall keep my judgments, and do them. And ye shall dwell in the land that I gave to your fathers; and ye shall be my people, and I will be your God. I will also save you from all your uncleannesses: and I will call for the corn, and will increase it, and lay no famine upon you. And I will multiply the fruit of the tree, and the increase of the field, that ye shall receive no more reproach of famine among the heathen. Then shall ye remember your own evil ways, and your doings that were not good, and shall lothe yourselves in your own sight for your iniquities and for your abominations. Not for your sakes do I this, saith the Lord GOD, be it known unto you: be ashamed and confounded for your own ways, O house of Israel. Thus saith the Lord GOD; In the day that I shall have cleansed you from all your iniquities I will also cause you to dwell in the cities, and the wastes shall be builded. And the desolate land shall be tilled, whereas it lay desolate in the sight of all that passed by. And they shall say, This land that was desolate is become like the garden of Eden; and the waste and desolate and ruined cities are become fenced, and are inhabited.

In this passage, the Israelites are given a new spirit and a new heart when they are forgiven so that they can dwell in the land and be blessed there. Their entrance into the kingdom required a "new birth" though it is not called that in the Ezekiel passage.

Similar statements can be found in Deuteronomy 30:6And the LORD thy God will circumcise thine heart, and the heart of thy seed, to love the LORD thy God with all thine heart, and with all thy soul, that thou mayest live.
See All... and Jeremiah 31:31-34 [31] Behold, the days come, saith the LORD, that I will make a new covenant with the house of Israel, and with the house of Judah: [32] Not according to the covenant that I made with their fathers in the day that I took them by the hand to bring them out of the land of Egypt; which my covenant they brake, although I was an husband unto them, saith the LORD: [33] But this shall be the covenant that I will make with the house of Israel; After those days, saith the LORD, I will put my law in their inward parts, and write it in their hearts; and will be their God, and they shall be my people. [34] And they shall teach no more every man his neighbour, and every man his brother, saying, Know the LORD: for they shall all know me, from the least of them unto the greatest of them, saith the LORD: for I will forgive their iniquity, and I will remember their sin no more.
See All.... Israel is to be as a nation "born at once" when the earth is made to "bring forth in one day" (Isaiah 66:8Who hath heard such a thing? who hath seen such things? Shall the earth be made to bring forth in one day? or shall a nation be born at once? for as soon as Zion travailed, she brought forth her children.
See All...). Nicodemus may not have been familiar with the phrase, born again, but he should have understood the concept. Then, when Jesus began to speak of the new birth, he should have been able to put together what He was talking about. However, he did not understand--not because the Old Testament said nothing of this new birth but because he and his compatriots were blinded to the truths contained there. He was a master in Israel (a master is a teacher as in Malachi 2:12The LORD will cut off the man that doeth this, the master and the scholar, out of the tabernacles of Jacob, and him that offereth an offering unto the LORD of hosts.
See All...), but he did not know the spiritual basics of knowing God.