As a recovering Catholic, I don't think in terms of the Bible much, but thanks to Ashley Long from createtheleap.com, I became aware of a book by Herbert Puryear titled "Why Jesus Taught Reincarnation". I started researching and was reminded that the Church used to include reincarnation in its teachings. In fact, according to Kevin Williams, 

"When Jesus began his ministry, many people wondered if he was the reincarnation of one of the prophets. Some people wondered the same thing concerning John the Baptist. And even Jesus affirmed to his disciples that John the Baptist was indeed the reincarnation of the prophet Elijah." 

"For all the prophets and the law have prophesied until John. And if you are willing to receive it, he is Elijah who was to come." (Matthew 11:13-14)

"And the disciples asked him, saying, 'Why then do the scribes say that Elijah must come first?'

"But he answered them and said, 'Elijah indeed is to come and will restore all things. But I say to you that Elijah has come already, and they did not know him, but did to him whatever they wished. So also shall the Son of Man suffer at their hand.'

"Then the disciples understood that he had spoken of John the Baptist." (Matthew 17:10-13)

This I did not know. The first quote on this blog is from Origen, and I learned more about him.

"The first great Father of the early Orthodox Church was Origen (185-254 AD) who was the first person since Paul to develop a system of theology around the teachings of Jesus. Origen was an ardent defender of pre-existence and reincarnation. Pre-existence is the religious concept of the soul as not being created at birth; rather the soul existed before birth in heaven or in a past life on Earth. Origen taught that pre-existence is found in Hebrew scriptures and the teachings of Jesus."

I'd never heard of the concept of pre-existence, but it resonates with my idea of the continuity of our consciousness before and after our physical incarnations. 

"The doctrines of pre-existence and reincarnation existed as secret teachings of Jesus until they were declared a heresy by the Roman Church in 553 A.D at the Second Council of Constantinople...(I)n 553 A.D., reincarnation became an enemy concept to the Judeo-Christian West. The reason reincarnation was declared heresy was given by Gregory, the Bishop of Nyssa. The five reasons he gave were:

1. It seems to minimize Christian salvation. (Inconsistent with the idea of eternal damnation in hell)

2. It is in conflict with the resurrection of the body. (Inconsistent with the rapture and final judgment)

3. It creates an unnatural separation between body and soul. ("For thousands of years, Christians believed that when a person dies their soul would sleep in the grave along with their corpse" until Judgment Day)

4. It is built on a much too speculative use of Christian scriptures. (Because there many quotes from Scripture referencing reincarnation?)

5. There is no recollection of previous lives. (Until now....)

All Hebrew and Christian scriptures support reincarnation: the Bible, the Dead Sea Scrolls, the Christian Gnostic gospels, the Torah, the Hebrew Bible, the Apocrypha, the Kabbalah and Zohar. 

http://www.celestialgrace.org/bible/Reincarnation_and_the_Bible.pdf

Thanks to Kevin Williams for his research.