Just so its clear, I'd like 'filth' as an evaporating flow that leaves behind trackable, disgusting grime on the tile. Outside, filth is washed away by rain and dwarves will try to 'use a shrub' if no options are possible. Once you start digging deep, problems arise.
Dwarves who do their smelly business randomly through the fort because you have no sanitation system get unhappy thoughts. Grime gets on their clothes and into wounds, causing infections. Dwarves have to clean themselves as well as the floors of grime, which causes more unhappy thoughts. The solution is to build a latrine, a 1x1 tile room consisting of a chair, block and bucket, using the 'o' tile. It is built over open space just like a well. You designate it into a room, assigning an owner if you wish or leaving it public. Using a latrine room when more than one dwarf is present in the room causes an unhappy thought; don't forget the door!
When used, the 1/7 bit of filth falls down into whatever hole exists to either splatter at the base of a tower, off a cliff, into a stream, into a cistern, or into a flowing sewer tunnel. In any case, doing so causes a happy thought (a golden throne and platinum bucket causing even happier ones). The dwarf is filth and grime free and returns to his daily routine of vomiting and getting covered in antman ichor. If filth doesn't begin to pile up, it evaporates into grime which can be ignored. If it does pile up, however, it spreads to other tiles using magma physics; unpressured and slow. A busy fort might find itself filling smaller cisterns. A 4/7 tile of filth generates miasma, which can rise through the latrine or any other openings.
Dealing with the filth production of a large fort requires water or magma. Magma obviously burns filth into smoke and causes potentially hillarious results, since a stream of flammable liquid trails from the magma all the way to your dwarf's behind sitting on a latrine. Clever players can come up with a solution or ten to this possible source of fun; floodgates, for example. However water behaves with filth differently. When water comes in contact with filth, any amount of filth, it is turned into filthy water and the two liquids combine. Filthy water leaves behind grime rather than mud, making it unsuitable for farming and bathing, and contaminates wells and pools. Dwarves must cautiously direct the water off site, into rivers or caverns, and possibly have evaporated away in the boundless magma sea. However, filthy water does not generate miasma or burn. It does, however, create an inviting habitat for a variety of mutated crap. I mean carp. Also tentacled beasties and kobolds, naturally.
Some people are worried that huge sewer systems will be required for forts; they are not. They are only required for huge forts. Your founders will be quite content to use shrubs and the like during the early months and once a latrine can be set up in a corner with a decent cistern below, you really have nothing to worry about for several waves of migrants. Once enough migrants amass that your latrines begin to fill and generate miasma, either dump the latrines into rivers, flush them with pumped or bucketed water, or seal up the latrine and dig newer, bigger ones. Only when your sprawling megafortress begins generating a new tile of 7/7 filth every week do you need the sort of sewer system that so many players long to explore in adventurer mode or defend against ratmen invaders. Because there is a lot of build-up time before the major demand arrives, most people should be able to agree this is a way to add fun and challenge to the end game. Kings demanding private golden latrines and sewer monsters crawling out of the deep? Sieges forcing us to close off our sewer system and disease running rampant through the fort? Yes please.
And of course, filth could be disabled in the init raw, and by being handled as 'filth' rather than specifically urine and feces, we put a fig leaf on the matter for the squeamish.