To clarify, I am specifically curious to know whether the period spanning the 6th to 14th centuries was notably different in its social organisation and in its economy from the societies which preceded Roman Hegemony. Was monarchy the original norm or was it inspired by Imperial Rome's legacy? Were European economies and societies as dependent on agriculture before 5th century climate change (approximately coincident with waning roman influence in the West) as they were after it?
OK, this is a huge question about which several forests worth of books have been written, but I'll give it a go. For the record, I'm a medievalist, so the focus in the answer will be on the post-Roman world, rather than pre-Roman Europe, about which I know much less. If there's any areas of particular interest you have, let me know.
First thing's first, the term 'feudalism' is particularly unhelpful in this context, so we'll ditch that. Have a look in the FAQs, or in my post history, for more reading than you could ever want about the difficulty of using the F-word.
With regards to your main question, the short answer is lots. While all pre-modern societies were heavily dependent on agriculture - being the primary occupation of c. 90% of the population throughout most of the Middle Ages, for example - the organisation of both society and the economy were heavily influenced by the Roman Empire. The administrative divisions of the Empire continued as counties and dioceses, which were formed out of the old city-territories (in some cases these city-territories themselves were continuations of pre-Roman divisions, however). The institutional church, which profoundly shaped the medieval world in virtually every aspect, was very much a Roman institution. The monarchies of the post-Roman west were initially based of Germanic kingship, but increasingly drew on Roman ideas of Empire and on Christian ideas of kingship, culminating in the crowning of Charlemagne in Rome in 800 and the foundation of the medieval German Empire (often anachronistically termed the Holy Roman Empire). The legal system in post-Roman Europe was essentially a continuation of the late Roman legal system, and the legal unfreedom which bound a large portion of Europe's peasants to their masters was the residue of the Roman slave system.
Like I said, it's a huge question, but the short answer is that things certainly didn't just return to the way they were pre-Rome.
Further reading:
Chris Wickham, The Inheritance of Rome and Framing the Early Middle Ages - The former is a nice general overview of the continuities and discontinuities of the Early Middle Ages after the fall of Rome, the latter is the big thick book on the impact of Rome on the society and economy of the post-Roman world.
Julia M Smith, Europe After Rome - More cultural in focus, this is the best account of how Roman though and ideas persisted and were developed in the medieval kingdoms.
Peter Brown, The World of Late Antiquity - A bit older now, but this is the book that basically invented the field of Late Antiquity, which has been incredibly influential in highlighting the links between the Roman world and the Medieval one.