I'm looking for works that try to trace which ideas of the Greeks, particularly their philosophical ideas, actually passed into Rome and were well-known enough to continue into the medieval period prior to the rediscovery of Aristotle. Anyone have any recommendations? Obviously this is a pretty large question, so any more specific works or just an answer with a general overview would be appreciated too!
The place to start would be Scribes and Scholars: A Guide to the Transmission of Greek and Latin Literature. Just be sure to get the fourth edition (2013), as the book has been updated several times over the decades with new discoveries and research.
There may be more specific Greek texts that you're interested in that might have specific articles and, very occasionally, books about them. But Scribes and Scholars is the general overview and has the benefit of being reasonably affordable for an academic book.