They don't specifically reference poor sanitation and widespread illiteracy (why would they?), but yes, there are some hints at people being uneasy about the spread and influence of Christianity in Rome. I'm going to take your question literally here, and focus specifically on the city of Rome itself. Partly that's because my expertise is on the city and the western Empire, rather than the east, but also because that's where some of the best evidence for this sort of thing comes from.

One of the most interesting sources of these kinds of feelings is actually the writings of early Christians themselves. They often responded to things that pagans had said about their faith, and through them we get a sense of the opinions held by others at the time. For example, Augustine's City of God was written as a response to pagan accusations that the spread of Christianity had caused the Sack of Rome in 410, because the city was no longer protected by its ancestral gods. This quote (2.3) is particularly revealing:

But remember that, in recounting these things, I have still to address myself to ignorant men; so ignorant, indeed, as to give birth to the common saying, "Drought and Christianity go hand in hand." There are indeed some among them who are thoroughly well-educated men, and have a taste for history, in which the things I speak of are open to their observation; but in order to irritate the uneducated masses against us, they feign ignorance of these events, and do what they can to make the vulgar believe that those disasters, which in certain places and at certain times uniformly befall mankind, are the result of Christianity, which is being everywhere diffused, and is possessed of a renown and brilliancy which quite eclipse their own gods. Let them then, along with us, call to mind with what various and repeated disasters the prosperity of Rome was blighted, before ever Christ had come in the flesh, and before His name had been blazoned among the nations with that glory which they vainly grudge. Let them, if they can, defend their gods in this article, since they maintain that they worship them in order to be preserved from these disasters, which they now impute to us if they suffer in the least degree. For why did these gods permit the disasters I am to speak of to fall on their worshippers before the preaching of Christ's name offended them, and put an end to their sacrifices?

As you can see, Augustine is quite rightly pointing out that Rome was hardly spared disaster when the pagan gods were worshipped, even before Christ's incarnation, so Christianity cannot be to blame for their current misfortunes. He goes on to pick holes in the argument in detail, quoting Cato, Cicero, Plato and others.

We do also have some pagan sources, although they are slightly less well known. After Rome stopped being the political capital of the empire at the end of the 3rd century, the city was pretty much left in the hands of the senatorial aristocracy. These families had held power and influence for centuries, and the pagan priesthoods were a central part of their authority. This did not stop immediately after the conversion of Constantine, and a hard-core group of aristocratic pagans maintained the traditional cults in the city through the fourth century. Other families did convert, as it was better to follow the religion of the emperors if you wanted to advance your career, so there was an ever-shrinking pool from which priests could be drawn. This led to individuals holding multiple offices, which would have been unheard of earlier in the Roman period. Vettius Agorius Praetextatus is a particularly good example, as you can see from him tombstone. One of Praetextatus' friends, Quintus Aurelius Symmachus, was Urban Prefect when the emperor Valentinian removed the Altar of Victory from the Senate House. This altar had been put there by Augustus after the battle of Actium in 31BC, and for centuries had been the place where senators had pledged their allegiance to the emperors and had sacrificed to keep the goddess of Victory on the Romans' side. As Urban Prefect and representative of the senate, Symmachus wrote a letter to the emperor pleading for the altar's restoration. It's a heart-felt plea, and one that at times comes close to modern ideas about religious pluralism and cultural tolerance. Take this quote for example:

We ask, then, for peace for the gods of our fathers and of our country. It is just that all worship should be considered as one. We look on the same stars, the sky is common, the same world surrounds us. What difference does it make by what pains each seeks the truth? We cannot attain to so great a secret by one road.

Symmachus appeals to the illustrious history of Rome, and to the current emperors' own forefathers, who had allowed pagans to worship despite their own Christian faith. There is, again, a hint that the Christianization of the empire leaves it open to disaster and defeat - Symmachus says " Who is so friendly with the barbarians as not to require an Altar of Victory?"

Fortuitously, we also have a reply to Symmachus' letter from Ambrose, bishop of Milan and imperial advisor (it's on the same website). His counter-arguments are similar to Augustine's - that Rome had suffered defeats and disasters before Christianity, so why had the pagan gods not protected the city then? The arguments of Ambrose and Augustine obviously won out - traditional Roman religion was never fully restored to the city and the empire. Whether that caused the ultimate fall of the empire is a considerable matter for debate, and not really something I can cover here.

If you want to learn more, I recommend Alan Cameron's The Last Pagans of Rome and John Curran's Pagan City and Christian Capital.