I wanted to do the question for Mesoamerica, but alas our rules say no. And with the recent trend of medical related questions associated with the Mexica I thought I'd add another.
Aztec births were a highly ritualized affair involving spirituality and midwifery. When the umbilical cord was removed it was preserved and when a son reached adulthood he would bring his to a battle field and bury it. A daughter would bury hers next to the family's hearth. There is an illustrated book called the Florentine Codex, I believe, that describes most (if not all) practices in regards to birth and raising children.
[Edited for link] (http://www.wdl.org/en/item/10096/)