Some parts weren't originally. Irenaeus testifies that Matthew's Gospel account was originally written in Judea in the language of the Jews, which would make it Aramaic or (more likely) Hebrew. Linguistic experts have determined that at least the opening and closing chapters of Luke's Gospel account were originally written in Hebrew and then translated into Greek when the good doctor incorporated them into his work. The sayings of Yeshua were also originally given in Hebrew or Aramaic and translated into Greek for distribution in the Gospel accounts. The Revelation likewise shows signs of being originally written in Hebrew and then being translated into Greek by someone who lacked Sha'ul's polish with that language--which makes sense, considering who wrote it and the circumstances under which it was written.
For the rest, I'm quite sure that most of the epistles and significant parts of the Gospel accounts were composed in Greek, since that was the common language and the apostles were writing for their audience. However, it's equally true that the authors were Hebrew in culture and in most cases (possibly all cases except for Luke) had Hebrew and Aramaic as their first languages. Therefore, while we should certainly understand the literal language of the Greek, it remains useful to understand Hebraic expressions, concepts, hermeneutics, and cultural idiosyncrasies.
And certainly that becomes all the more important every time a NT author quotes or alludes to the Tanakh.
The NT authors wrote in Greek because they wrote to a pagan culture--however, they did not write from that culture, but from the holy culture that God Himself established at Sinai and shaped through the centuries, both in the Land and in the Diaspora.
Not exactly. The Irenaeus quote goes like this, "Matthew also issued a written Gospel among the Hebrews in their own dialect ("dialektos") while Peter and Paul were preaching at Rome and laying the foundations of the church."
The Greek word dialektos can mean language or style. His quote is most likely based on a earlier statement by Papias, "Matthew compiled the sayings ("logia") in a Hebrew dialektos and each one interpreted them as best he could."
Most scholars agree that the text we have today was mostly likely written in Greek, and not a translation from Aramaic. Papias could have been referring to the Hebrew style, or he may have had another work in mind, e.g., the "Gospel according to the Hebrews"
Linguistic experts have determined that at least the opening and closing chapters of Luke's Gospel account were originally written in Hebrew and then translated into Greek when the good doctor incorporated them into his work.
Do you have a reference? It seems odd that portions of a work directed to Theophilus the Greek would be written in Hebrew.
The sayings of Yeshua were also originally given in Hebrew or Aramaic and translated into Greek for distribution in the Gospel accounts.
That is most likely true.