Unless encountered at a major base, the taunt will not have musical accompaniment.
The meaning of the phrase "А ну, чики-брики и в дамки" which Cheeki Breeki is derived from, relates to Checkers and is also part of a Russian nursery rhyme. The phrase above translated phonetically to English, it is, "Ah nu cheeki breeki iv damke".
The misconception of the expression "cheeki breeki" says that it is a corrupted pronunciation of "shake it, break it", but the actual phrase itself has two varying translations, which both have a similar meaning,
А ну, чики-брики и в дамки = "One Two, you're on top!" To stack two pieces to create a king in checkers. Simply put, to gain the advantage against the other player.
Another translation depends on the Bandits heavy use of localized Russian slang. "чики-брики" is often warped into a vulgar threat of shooting someone, while "и в дамки" means "The Head" which would roughly translate the phrase into "Shoot him in the fucking head".
And beyond the files of the Bandits, they have different types of sound files that play when they try to communicate. For example, "A nu cheeki breeki iv damke!" is a "detour" file that plays when Bandits try to flank the player, like said in the name "detour", means to go around. Going back to checkers, roughly, it means to take advantage, combine this information with this "detour" file, and the popular phrase "A nu cheeki breeki iv damke!", means to "Flank from behind!"
But why do Bandits speak such incoherent phrases? By looking at their faction, we can already tell that they are a poor, low-class faction. They are wanted for their means of criminality, and are hated by all the other factions. So they make their own phrases so that the other factions cannot understand. To put this simply, they just try to confuse you, so they can "Flank from behind!"
The "Bandit Radio" track is also associated with Cheeki Breeki and is written in the genre of "Chanson" which is a genre listened to by Russian criminals, street-trash and the aforementioned Gopnik stereotype. Said track is based on an existing track called "Хоп, мусорок" which loosely translates as "Hop, little trash policeman".