Jesus Speaking Aramaic
There is no doubt that Jesus likely spoke Aramaic and that this language was pretty common among Jews in his day. This is confirmed by passages like the following:
Of course, in these passages Mark makes it clear that though he is writing in Greek, the original words of Jesus were spoken in Aramaic.
And taking him aside from the multitude privately, he put his fingers into his ears, and he spat and touched his tongue; 34 and looking up to heaven, he sighed, and said to him, “Ephphatha,” that is, “Be opened.” (Mark 7:33-34)
Greek in Ancient Palestinian Judaism
But it also seems that Greek was known to ancient Jews. The influence of Greek culture in the first century is clear from the fact that the high priest in 37 A.D. had a strikingly Greek name--Theophilus. It is striking to me that he did not feel the need to change his name to a Semitic one--there was apparently no problem with this high priest being called Theophilus (Ant. 18.123).
Many sources could be cited here. Stanley Porter writes,
The evidence Porter cites is pretty convincing if you read his whole treatment which includes an impressive bibliographic material in footnotes (omitted here).
". . . Jesus would probably be best described as productively multilingual in Greek and Aramaic, and possibly Hebrew, though Aramaic would have been his first language and Greek and Hebrew being second or acquired languages. . . He may also have been passively multilingual in Latin, although if he had any knowledge of Latin at all it is likely that it was confined to recognition of a few common words. This depiction reflects the linguistic realities of the Mediterranean world of Roman times, including that of the eastern Mediterranean, and is supported by widespread and significant literary, epigraphic, and other evidence. As a result of the conquests of Alexander III ('the Great'), and the rule of the Hellenistic kings (the Diadochi and their successors), the Greco-Roman world was one in which Greek became the language of trade, commerce and communication among the now joined (if not always united) people groups. In other words, Greek was the lingua franca for the eastern Mediterranean world, displacing Aramaic. . . .
The arguments for the use of Greek in Palestine are based upon the role of Greek as the lingua franca of the Roman empire, the specific Hellenized linguistic and cultural character of lower Galilee surrounded by the cities of the Decapolis, and the linguistic fact that the New Testament has been transmitted in Greek from its earliest documents. There is also a range of incriptional evidence (e.g., Jewish funerary inscriptions), numerous Greek papyri, and significant literary evidence, including Jewish books being written in or translated into Greek in Palestine. From this range of evidence, the logical conclusion can be drawn that in fact a sizeable number of Jews in Palestine used Greek."--Stanley E. Porter, The Criteria for Authenticity in Historical-Jesus Research: Previous Discussion and New Proposals (London: T & T Clark, 2000), 134-5, 140-141.
The Original Aramaic Substratum
I have no doubt that Jesus typically spoke Aramaic. However, scholars often get hung up trying to discover the "Aramaic substratum" behind the Gospels' accounts of Jesus' words. This seems problematic. It appears possible to me that there may have been no "original Aramaic" in some cases.