After getting picked up by hovercraft as described in the earlier answers, the bodies get shipped back to the districts and the mourning families. I haven't got the book here to back this up with a quote, but in 'Catching Fire' Katniss thinks about what will happen to her when she's dead. Peeta's just given her the pearl, and she thinks something like "surely they will know to leave it with my body when they send me back in a box to my mother and Prim". (This is right at the end of a late chapter, if someone would like to edit with the relevant quote!)

After being picked up by hovercraft as described in @Solemnity's answer, the bodies get shipped back to the districts and the mourning families:

I have the pearl, though, secured in a parachute with the spile and the medicine at my waist. I hope it makes it back to District 12.

Surely my mother and Prim will know to return it to Peeta before they bury my body.

^{-- Catching Fire, end of chapter 25}