The Etruscans assimilated into the Roman Empire, so Italians still in that area are probably descendants. We still have Etruscan-style art and jewelry being produced.

I believe the left over Olmec people became Mayans when they're numbers came up, but not sure.

There has not been a real "mass extinction" of people since pre-civilization times if that's what you mean.

The Easter Island (Rapa Nui) people are still around speaking their native language and all that. They suffered some severe population crises, two of which were caused by Europeans, but they are still living on the island.

It's a bit like speciation and evolution in biology. When we talk about a civilization like the Mayans or the ancient Romans, we define it based on shared practices, language, and cultural production within a certain time and place. But these criteria are determined long after the fact by historians and archeologists, for the purposes of study and categorization. In reality, cultures gradually morph over time, and it falls to the scientists to draw the line between one ancient culture and the next one. Dinosaurs evolved continuously into birds, yet we still think of them as extinct, because at a certain point in the fossil record, they start looking too much like birds to be called dinosaurs. Likewise, Roman and mayan culture did not vanish in a flash, they just continued to evolve.

The term 'civilisation' is a bit contentious in itself, and not always easily defined. Its one of those thing that we know what mean, but definition can be fraught with exceptions and margins. In the Mayans case its generally accepted that the people in the region were a mix of farmers/hunters/gatherers who at some stage developed into a civilisation based on their ability to build and organise cities and a socioeconomic stucture based on and and supporting those cities. The Mayan civilisation seems to have fallen around 900AD. We dont know nor can we be specific about the date. It occurred when they Mayans largely abandoned their cities and ceased building new buildings. They would then have largely returned to the land. The collapse appears fairly quick near the end and its probable that a person born near the end would have been raised in a city and died in a village. But the people, the language and the beliefs continued on. Its just the cities, one of the primary indicators of a 'civilisation' (except for the Mongols), that came and then went.

Whenever I and many other scholars use the word civilization, we do not refer the actual people. It's an easy short cut to refer to the certain complex of culture, power power, economic activity, etc. of a particular area and period. When we say a civilization "no longer exists", it's because one or more of those elements have collapsed, been replaced with a foreign one, or evolved beyond recognition. The people, though, remain.

Many sites in Peru are excellent examples of this, e.g. Pachacamac, Chavin de Huantar, and the site I'm currently working at in the northern highlands. First, a general chronology to help: ~800-200 BC, the Chavin religious cult is present throughout the Andes. 200 BC - 600 AD, the "Early Intermediate Period" sees unique regional cultures (e.g. Moche, Recuay). 600 AD - 1000 AD the Wari empire controls much of the northern Andes, and Tiwanaku is dominant in the south. 1000 - 1400, the "Late Intermediate Period" again sees local cultures that will eventually be assimilated by the Inca and Spanish. Depending on your area, that's 4-6 "civilizations that have come and gone. At my site, there are construction phases from each of the these periods, and each one coincides to a "lost" civilization. The people are still living in the same spot but they are interacting with the dominant culture or political power in a different way. The Chavin appropriated an older one, the Huaras culture formed of the remnants when Chavin lost it influence. Recuay chiefs brought definitive political power and introduced new ceremonial spaces. The Wari imperial didn't reach too far north, but it's ceramic style became the hip thing to use. The cult center of Chavin de Huantar might provide the best example of this: Recuay households have been excavated right in the middle of one of the center's ceremonial plazas, using old relief carvings in its walls. The people are still there, but their relation to/use of certain cultural symbols and political powers changes.