Most states do not require
embalming.
--Funeral homes may require it if you have a viewing.
--Airlines may require it if you need to transport the deceased by air.
Some states require it
only if the body is not buried within a certain time frame (example: Minnesota requires it if the body is held more than
72 hours, not 24) or sometimes if the person died of a communicable disease.
Most states do not require
caskets.
--some
cemeteries do require it
--even cardboard is an acceptable casket, in fact many bodies are buried in cardboard boxes.
Most states do not require
vaults or grave liners.
--
cemeteries usually do require these due to settling of the ground when caskets decompose. (It is not a law, just a requirement of each cemetery, even in the new jersey case mentioned upthread)
Green burial is allowed pretty much in every state, it's the cemeteries that may not allow them. Most states allow the family to bury their own dead with few regulations, actually, the only real limiting factor is finding the ground/land to do it. If you have land in the country, you can pretty much go for it.
My good friend died last year and we buried him on his farm. Legally. No previous cemetery was there, he's the first and only body. No vault, no real casket (he's in a simple unfinished pine box, think plywood). The only regulations we had to deal with (Illinois) were that he had to be buried 18 inches below the surface and he had to have some sort of permanent grave marker within 1 year. We had a local artist friend build a stone bench and made a sculpture-like pillar with clay tablets with his name and some words, and with a gorgeous pottery bowl set on the top.
We also, like most in this thread, had no idea this was possible (burying him ourselves at home) but after he died, it was actually suggested to us by the Unitarian minister (not sure they call him that, I'm not UU). He had already been taken to a funeral home by the time it was decided and they assisted us with finding out the legalities. Turns out, there's just not that many legalities involved.

I guess it's pretty easy to assume that the normal way to do things is the only way, or is done because it's the law and you have to do it. In the end, people do what the cemetery requires, and what the funeral home requires because they believe it's state law. Most often, it's not.