> how much of the Hittite cultural aspects has been carried into Greek
thought?
The timing is somewhat against that idea, though not totally. The Hittite
empire comes to an abrupt end around 1200BC, as Greece enters a cultural
dark age, when much of the previous cultural and social structures,
including writing, were lost. Greece emerges from this in the 8th century.
Partly that is due to the re-establishment of trade and influence from the
middle east. (There is even a style of pottery called "orientalising".) It
is not clear precisely which Eastern cultural group was the source of this
influence. The Phoenicians are a strong candidate, certainly for
transmitting it. But the Hittites disappeared 400-500 years earlier.
There are two possible sources of influence though. One is contact between
Mycenae and the Hittites. It is at least possible that some Hittite ideas
could have been picked up, and might have been transmitted within Greece for
400 years. I don't know of any evidence for this though. Secondly, we know
that there was strong contact between Greece and other Anatolian groups,
such as the Carians and Lycians and Lydians, some of whom might have
conveyed some Hittite ideas into Greece - possibly.
So the simple answer is "probably not very much, if any".
Peter