In most depictions of Medieval times, people are pictured as destitute and filthy; in contrast, most depictions of Antiquity - such as Rome and Greece - show the people being clean and orderly. Were people generally cleaner back then?

I don't have any sources that discuss this issue in specific, so I've just pulled together a few bits and pieces for you regarding Rome, primarily the city itself.

Rome's sewerage system was impressive in scale and sophistication, but did not completely translate to a totally clean city (Carcopino 52). Individual houses often did not have access to these sewers, as one might assume, but rather the system mostly collected sewerage on the city floor and from public latrines, though the very rich may be able to build a latrine directly over the aqueduct (52). The poor, on the other hand, had to go outside to 'cleanly' relieve themselves, though the public latrines were very cheap and easily accessible (52-3). The landlords of the city of Rome did at least have a responsibility to keep the public street side of their buildings clean, on pain of a significant fine, though this was not necessarily well police (58-9). I should note that some respectable commentators, including one I found by the name of Ann Olga Koloski-Ostrow, have suggested that the presence of sewers and public latrines did not necessarily translated to a very clean city, and thinking of the limitations of their hygiene knowledge (clean hands?) and system this perhaps isn't surprising.

Public baths would have been helpful, while rich Romans had their own bathing halls even the poor could access (277, 285). Carcopino suggests that physical cleanliness was an ideal that the Romans pursued in the public baths, though I hesitate to take this idea further (286). The Roman must have valued cleanliness however, because they were also quite prudish and the public baths certainly attracted criticism for the potential lasciviousness that could occur within, but they remained very popular. Roman men of all shades tended to bathe in the afternoon, but not in the morning (174). That daily bathing was common however is certainly an argument in the favour of a fair standard of contemporary cleanliness for the average Roman.

Rome had an incredible abundance of resources and logistical organisation that allowed it to feed and water its population, especially in the city of Rome itself, that would have been very helpful for the health of its inhabitants (Patterson 356-60). To qualify, while the city had enough water for its inhabitants, but this did not translate to private households having a running supply- individuals needed to collect their water from public fountains (Carcopino 49-51). That said, in many places the huge population of the city and small spaces between many buildings also meant it was very crowded, meaning disease could spread quickly (like fire and bad news). But as we know diet and water and essential components to the good health of humans, these were strong factors in Rome's favour that nations in other places and times with less resources may not be able to compare to.

I will leave it to somebody else to compare between Medieval and Roman Europe, as I know little of the former, though I will qualify that by pointing out that many of the hygiene-related structures that were built in Rome in the late Republic and Empire such as "drains, aqueducts, public gardens, green spaces and baths" were a product of Rome's uniquely massive resources and the advent of one-man rule where individual rulers desired to show their beneficence to the people (Holleran 285-6). Only a rich and long-established state can build such structures in their cities, and I daresay that most of Medieval Europe lacked this capacity for much of the Medieval period.

I might say in general that the Medieval period is 'popularly' viewed as dirty, rustic, disorganised and the classical period, especially the Romans and Greeks, as being quite the opposite. I think in general, and I'm sure most Medieval experts would agree however, that this is in large part a product of a general tendency to criticise the 'Dark Ages' and glorify the Ancients, as has been fashionable ever since the Renaissance. Rome certainly possessed incredible physical resources and organisation that its successor states in Europe could not compete with for a long time, but you don't have to look far to see examples of European states that do possess great resources and organisation (such as the Carolingians). I would not be surprised to find comparable examples of hygiene in the great cities of the most powerful Medieval European states. After all, people weren't totally clueless on hygiene: its importance was understood by armies on campaign regarding latrines and water (deliberately fouling water supplies was a very old tactic), so surely it was understood to some extent by city authorities of any great power.

Sources:

Carcopino, Jerome. Daily Life in Ancient Rome. Penguin Books, 1947.

Holleran, Claire. Book Review: Rome, Pollution and Properity. Dirt, Disease and Hygiene in the Eternal City from Antiquity to Modernity. The Classical Review, 64, 2014, 285-7.

Patterson, John R. "The City of Rome" from A Companion to the Roman Republic, eds. Nathan Rosenstein and Robert Morstein-Marx, Oxford: Blackwell, 2010