Only primitive forms of filtration were practiced - ie running extremely disturbed water through a sieve or the like - cloth or unglazed earthenware. But this was small scale in any case, and typically only removed the worst impurities.

Much of the clean water delivery from Roman technology wasn't by cleaning the water itself, but by finding a clean source and delivering it through a clean channel - they would get as near to the spring as practical for a reservoir, and then have that feed into an all stone/concrete aqueduct. Since there's nothing 'dirty' to contaminate the water, then, unlike in the course of a river's flow, the water is delivered clean over however many miles it is to the destination. And that was considered the 'easiest' way to get clean water!

The Romans used settling basins to ensure that some turbidity that slipped in anyway was removed, but that's just giving the water enough still-time (and a deep enough basin) for gravity to work its magic, letting heavier solid sink beneath the channel's next opening before sending the water onwards.

The Mayans, interestingly enough, appear to have developed a form of sand filtration, and using extremely excellent materials for the job at that, but such techniques are absent from the rest of the world, to my knowledge, until ~1804 AD.

Insane, right?

London itself wouldn't receive mandatory water filtration until 1852, and that was considered a cutting-edge move.

Part of the reason why it's non-intuitive is that slow sand filters are extremely effective because they perform their function by two processes. Some turbidity is removed by how it's 'sieved' through the sand and the rocks, but much of the bacteria and parasites are actually removed by larger microorganisms in the sand. The larger microorganisms mostly can't filter down more than a few inches through the sand before they suffocate or the like; the smaller microorganisms present in the water that could filter through are slowed for long enough that they're devoured by the larger ones.

The circle of life!

However, part of it is also just that... many ideas we take for granted are extremely modern inventions. Hell, doors didn't have knobs until the 18th century, just handles. A simple idea is often worth a person's weight in gold. "Let's use sand and gravel to make water cleaner!" Congrats, you just saved millions of lives.

Ancient peoples knew stagnant water was generally unsafe, and that moving water was generally safe; and likewise that clear water was generally safe and that turbid waters were generally unsafe. However, that was as far as the understanding went - if the water tasted bad, that was another warning sign, but they had very little conception of germ theory. If the water looks clear and tastes fine, how can it be bad?

While clear water without a foul taste is a better sign than the reverse, it's not nearly sufficient to guarantee that the water is healthy - bacteria can be present in large amounts before it starts to leave any sort of taste. So water filtration was not a high priority, and even if it was... well, no one had the idea (except the Mayans) to try to recreate the process of rain becoming groundwater before 1804.