Today you can find various books written in Syriac or teaching Syriac or translated from Syriac, in which the author claims that this is "Aramaic, the language of Jesus." Well, it is true that Syriac is one of the many dialects of Aramaic, and it is true that Jesus spoke Aramaic. However... (sorry to burst any bubbles) Jesus did not speak Syriac. Jesus spoke a rather different dialect of Aramaic. The speakers of his dialect could probably understand Syriac, and vice versa, but there may have been some difficulties in communication. So when you see a book that advertises Syriac as the language of Jesus, take this claim with a grain of salt. Even so, by reading works in Syriac, especially ones written before the Council of Chalcedon (451 CE), you will be reading a language whose thought categories are much closer to those of Jesus than those of Byzantine Greek which later became dominant in theological discussion (particularly during the Christological controversies of the 5th century, which is why I give the Council of Chalcedon as a dividing point). The Syriac writings of St. Ephrem of Nisibis are the epitome of Syriac literature, but they also take the most study to fully understand.
Although I have cursorily mentioned Greek influence above, I should be more specific. The Syriac-speaking area was by no means untouched by Greek. Syriac was the dialect of Aramaic spoken in the city of Urhay, and by the time Ephrem was born in the nearby city of Nisibis in 303 CE, Edessa had been under Greek influence for some six centuries, thanks to the conquests of Alexander the Great. Ephrem himself was quite knowledgeable of at least one school of Greek philosophy, and the Syriac language had quite a few Greek loanwords (quite a few Persian ones, as well, but that's another story). Even so, Ephrem had a talent for expressing himself in a rather traditional Aramaic-sounding way, so his writings are a rich source of insight into Aramaic thought categories in the 4th century of the Christian era.