Well, no. Not really.

While medieval cities were a far cry from what we would call "clean", they did at least try. We do know that people on occasion did toss garbage, offal, and human waste into the streets because there are numerous edicts and laws passed throughout the Middle Ages where individual cities would outlaw such practices, often leveraging fines on the offenders. For example, London instituted fines for leaving excrement (human or animal) in the roads in 1309, further rules against where you could and could not throw refuse in 1357, and the 1388 Statute of Cambridge set the fine for dumping excrement, trash, or offal in ditches and waterways at a whopping £20.

Most houses in the cities, e.g., London, had access to either a cesspit or a privy. These would need to be cleaned out on occasion. This, combined with needing to clean the streets of gunk, gave rise to a not terribly prestigious but nonetheless necessary occupational opportunity. In England these folks were called "Gongfermors" or "Gong farmers"--part garbageman, part street-sweeper, part sewage removal technician. Some enterprising individuals even made a decent amount of money.

People were sensitive about smells in the Middle Ages. It was tied to their ideas of cleanliness. Bad odors--specifically rotting offal and sewage--were thought to be a vector of disease. The reality of logistics prevented cities from getting rid of this waste entirely, but people were at least aware of the problem and did at least try to remedy it.

TL:DR: if someone in the Middle Ages in England did what your professor described, they'd likely have been slapped with a hefty fine.