By "classical" times, I'm going to assume you mean "the period in which what we call classical music existed" rather than "Classical Greece"!

Yes, folk music! "Folk" simply means "people", and folk music is the music of the average person of that ethnicity or nationality. It was often unwritten, with songs passed down through training and practice to learn by ear. Folk and classical music exist in parallel, with classical music descending from strict European traditions that originate in liturgical music.

Folk music also holds a very important role in being the originator for much modern music! Jazz, rock, country, and blues all originated as folk music, especially black residents of the southern United States who incorporated traditional African music tropes (like a call-and-response structure) to the new culture they were creating.

These forms of music grew in popularity to the point where they were eventually appropriated by white performers, starting with the popular "big band" ensembles and eventually moving into rock n' roll and country. And as we all know, the influence of these genres would extend in uncountable and (for some people) unfathomable ways. Metal and all of its subgenres, electronic music, modern pop with its R&B and electronic influence, etc. All of it is rooted in the centuries of folk music that eventually intermingled.

You can also see a very modern type of folk music: hip hop! Rapping comes all the way from traditional African spoken word chants and stories with musical accompaniment, with black artists in the early 20th century like the Memphis Jug Band and Joe Hill Louis using rap-like delivery. Combine that with influence from spoken word poetry and the black and Latino cultures it grew up in, and you now have one of the biggest musical genres in the world emerging organically from folk traditions over a very short period of time.