This is a somewhat controversial topic within the Syriac christian community. Basically there are two main schools of thought: those who identify as Assyrians and descendants from the ancient Assyrians and those who identify as Arameans and their descendants. I will let someone with the 'Assyrian' origin point of view post that since I'm not as well versed in their various arguments but I will provide you with the basic 'Aramean' origin arguments, however first I'm going to go on a slight tangent that is necessary for this topic. Up till the 20th century, the Syriac Christians called themselves in Arabic 'Suryani' which originally meant 'Syrian'. Suryani was not only a term exclusive to Christians but also exclusive to those Christians not of Arab ancestry (the arabs christians were usually Greek Orthodox, Greek Catholic, or Maronite). During the 20th century however, nationalist ideals started among the Arab Christians of 'Al-Sham'(the Arab term for the Levant meaning "land of the North"), and at the helm of this nationalist movement was Antun Saddeh, a Greek Orthodox arab christian. In his movement, the arab christians who identified as 'al-shami' (meaning arab of al-sham), started to adopt the name 'suri' an invented term that was equated with Syrian in English therefore causing great confusion. The 'Suryani' response was to change the translation from Syrian to Syriac. So for the purposes of this post I will be using Syriac, and take in mind that it means 'Suryani', unless I specify that I mean the Syriac language.

The main idea of the Aramean origin argument goes as this: The Syriac christians, are descendants of the people called Arameans, a group of northwestern semites closely related to the Hebrew and Phoenicians, who during the bronze age collapse settled the land they called Aram. Later on, due to outside influences, namely Greek confusion on the matters of the people they saw as barbarians as they saw all non-Greeks, the name 'Syria' was invented either from a)the Coptic word for Hurrians translated to Greek or b)the Luwian-Hittite word for Assyrians translated to Greek (note that despite there being a greek word for Syrian, the Greeks also had a word for Assyrian, therefore complicating the matter even further). During this confusion the Greeks and later on the Romans would use the name 'Syria' without any solid meaning to refer to people of the near east/fertile crescent region who spoke aramaic. However, during that time, the Arameans continued to identify as Arameans until a gradual process started that was most likely in tandem with the rise of Christianity(which was after all spread by Hellenized Jews who spoke Greek), where they started to called themselves 'Syriac' in their native Aramaic. Eventually, the term 'Syriac' would be used to apply exclusively to the people of Aramean ancestry. So the name Syria would be applied to the former lands of Aram, which makes sense since the Syriac Church was based in Antioch, a city nowhere near ancient Assyria. So due to the fact that Arameans adopted the name 'Syriac', anyone who identified as such was an Aramean whether or not they came from areas like Mosul or Nineveh which thousands of years earlier were considered the Assyrian heartland. They back this up with evidence that during the Iron Age, Arameans settled in large numbers in the Levant and a region of Mesopotamia called Aram-Nahrain (Nahrain meaning "of the rivers", possibly a callback to the bronze age kingdom of Mitanni which was from the same area of Mesopotamia, that was called Nahrain by some peoples), which was just north of the Assyrians. The Assyrians on several occasions held massive deportations of Arameans (as well as other peoples such as Jews), not out of the empire, but into its capital. It is claimed that Aramean numbers grew so large that Aramaic replaced Akkadian, and Arameans grew to outnumber the native Assyrians. The origin of the modern Assyrian identity would be in the works of 17th/18th(not sure which one) century western missionaries who upon finding the Syriac christian communities of Iraq, claimed based on archeological grounds that they must be the descendants of the ancient Assyrian, confusing the meaning of 'Suryani'.

Let me know if anything needs clarification, however, I will restate that I expect someone else to post the Assyrian side of this, as I do not want to misrepresent my post as one sided.

There's a mod on here that supports the continuation of the modern Assyrians from the ancient Assyrians: /u/Daeres

Here are some of his excellent posts:

http://www.reddit.com/r/AskHistorians/comments/17h4d9/what_was_the_significance_of_the_name_change_of/

http://www.reddit.com/r/AskHistorians/comments/10qocs/were_the_assyrians_victim_to_a_culturewide/

http://www.reddit.com/r/AskHistorians/comments/10m94x/friday_freeforall_sept_28_2012/c6eo627

Here's a quote from the last thread:

"One thing to bear in mind is that even before Assyria became an Empire, it was the largest state in the Near East. Many of its capitals were destroyed when the Empire was toppled, but several of its major centres continued to be active. Even reduced in size and strength, and without political control, there were still a lot of Assyrians- it was a state capable of raising a standing army of 80,000 Assyrians by about 900-800 BC. We know that worship of Ashur at the city of Assur was still going on before the Arab conquest. That means that the Assyrian identity had survived for 1200 years after the fall of their Empire. It's now about 1400 years since then.

Given how many cultures have assimilated or vanished in that 2600 years, I'm just grateful that Assyrians aren't gone from the world."

And here's a great post about the Arameans from another historian:

http://www.reddit.com/r/AskHistorians/comments/2zcvnq/what_is_the_history_of_the_arameans_when_was/

Short post about "Syrian vs. Assyrian":

http://www.reddit.com/r/AskHistorians/comments/2u5p7f/is_there_a_difference_between_assyrian_and_syriac/

Edit: Fixing a link.