Speaking from a biologists point of view, the cleaning off of utensils just by running water and a soft scrub is enough to prevent growth of most bacteria on a metal or wooden surface. Even if there are bacteria that remain (which there will be since there are bacteria everywhere), the chance of someone getting sick from that is minuscule. Food borne illness is more likely to come from improper food preparation than the plate or utensils used. The human body is very good at preventing and fighting bacteria.

Now in terms of food handling techniques, can you be more specific on which civilization? Both food preparation and food handling are highly specific based on the culture and time period. For example, the religious laws of Judaism are very specific on how (and which) food is to be prepared and consumed. However, the Greeks, Assyrians, and other cultures also had specific ways of preparing food. As an example source. Now this source is about the ethics of food consumptions, but it also contains some specific examples.

According to the book, A History of the World in 6 Glasses, the discovery of the fermentation process, coupled with realizing that drinking alcohol (beers and wine early on) were safer than drinking water (as those who drank water became very ill) propelled early civilizations to not only extend the process of fermentation, but also to use it as a method of payment In ancient Babylonia and other parts of the middle east, a workers daily payment included fresh bread and a set amount of beer. On a more modern note, George Washington was looked down upon by many European leaders (after the American Revolution) because he would toast with water. Most Europeans drank wine for the aforementioned reasons, whereas in the states, many of the water sources were still clean enough to drink from without fear of getting sick at that time. I should also mention that GW was a brewmaster and knew how to make some kick-ass beer.

You must remember that they had no concept of germ theory or the existence of bacteria. That came from Pasteur.

They washed things to get rid of dirt and grease. They did not wash hands after they used the loo unless they obviously got something on them. Almost everyone cleaned their hands before eating, but in cold water without soap. (Rickert, The Babees' Book; Hale, The Horizon History of Eating and Drinking; Visser, Dinner is Served)

They survived by surviving. Those with weak immunosystems died younger (check the high infant mortality). Also check the frequency of great plagues.