No known music pieces, because Vikings had no musical notation. But we have plenty of archaeologically recovered instruments such as horns (literal modified animal horns as well as wooden), bone flutes, wood flutes, wooden panpipes, a metal percussion instrument called "jaw harp", a piece that could've been part of a bagpipe-type instrument, plus more conventional lyre-style instruments, drums, etc.

All these can be reconstructed by experimental archaeologists and played, so we definitely know what they sounded like.

In their own (later, Christian-era Latin) texts, lyre instruments are mentioned a lot and seem to be the most popular. And no discussion would be complete without this account from Arabic traveler Ibrahim Ibn Ahmad Al-Tartushi:

”Never before I have heard uglier songs than those of the Vikings in Slesvig (in Denmark). The growling sound coming from their throats reminds me of dogs howling, only more untamed.”

Which is fascinating when you think about the thriving death, black and folk metal scenes in Scandinavian countries and the fact that they pioneered these genres! I'd love to give the likes of Insomnium or Amon Amarth some reconstruction Viking instruments and see what kind of songs they make!

There is some folk music inside the borders of present day Norway that possibly existed in the Viking era:

• Sami joik is believed by most to be thousands of years old, and some joiks MIGHT have existed since the Viking era and earlier.
• Finnish runo songs (not to be confused with runes) have been sung up until our time in the Finn forests in Østlandet and among the Kven in Finnmark and Nordland. These are part of a song tradition that is at least as old as the Viking age, some claim they are 5-6000 years old.
• Chants to herd animals - 'kulokk' being most famous - are impossible to date with precision, but they are certainly old and known from many cultures.
• Songs to calm babies are also very old, some of them we can date, but some are possibly as old as Viking age
• Children's songs - children have been known to be very good at carrying traditions through play. For example, the song Tre små kinesere på Højbro plads was created among German-speaking children in German colonial outposts in China, and the Chinese in the song originally didn't play the kontrabass (double bass), but lacked reisepass (travel documents) - which is why police came for them. That is a fairly new song compared to London Bridge is falling down or Bro, bro brille, which both are very old. Some claim Bro, bro brille predates Christianity and contains memories of human sacrifice, some say both songs have to do with the battle of London in 1014 when Olav den Heilage allegedly tore down London Bridge.
• Which brings us to Draumkvedet. We can not know how old Draumkvedet is, some say it is a modern invention, some say it is from the 12th century, others place it some place inbetween, though most believe it predates the reformation. The "bridge" image is important in it, a throwback to the Gjallarbru of Norse mythology? So, in both these children's songs and in Draumkvedet (and other ballads/songs) we can maybe find traces of pre-Christian Norwegian/Nordic/European culture.
• Short songs like stev probably came about around the 12th century, so some of those could possibly be that old.

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Olav Bø: Utsyn over norsk fokedikting

https://urn.nb.no/URN:NBN:no-nb_digibok_2008052104061

Henning Frederik Feilberg on Bro, Bro Brille
http://www.rimogremser.dk/brobrobrille/pdf/efterskrift.pdf

Different sources for Draumkvedet

https://www.dokpro.uio.no/ballader/lister/tsbalfa_titler/tittel_61b.html