DF is probably somewhere in the Medieval era. By that, the most realistic would be pots and throwing your... waste out into the streets or carrying it all the way to the garbage dump.
Or have someone else carry it there, like a good hauler.
Personally, I'd rather not have my dwarves continuously having to carry their dung outside the fort, polluting the whole environment.
You can always go the Roman route, and use aqueducts or rivers to flow water beneath public latrines to clean waste water away. But that would add some additional complication not everyone would be up for setting aside as a necessary side-project for a lively fort.
However, the prospect of having a dedicated giant water tower devoted to flushing the latrines every once in awhile does sound interesting, and could be like one of those roadside attractions (of the "World's largest thermometer" variety) for tourists to visit/migrate to your fort for.
Interestingly enough, this gives me an idea for a future feature I wonder if Toady will implement later. Tourism and Pilgrimages. There will be random chances of visitors of the non-trade variety to check out your fortress in all it's glory (especially of high value; you can try to charge them or not, using your accountant or something, to setup the prices), and if you have plenty of shrines to your gods setup (a masterwork/artifact table can be assigned as a shrine/altar, which can attract pilgrims to your fort (and you can extort them for what they're worth, charging them gold or offerings (from wine to blood; settings could be made by your head priest(ess)/pope), when the economy kicks in). Of course, your own dwarves which celebrate the same god(s) may also donate to the church/shrine as well. They can be put to the most use when bazaars return to the game (shops, shops everywhere/market districts). Your income doesn't have to come from traders alone, especially when economy returns.
EDIT:
Just realized, in case holy wars are included, sometimes setting up a shrine will invite hostiles from other nations that celebrate other gods/religions. So we don't only have race wars (gobbos and elves vs. dwarves; but sometimes other dwarves vs. other dwarves, and similar to other races). Could lead to some interesting history generations (watching numerous obscure religions eventually monopolize the belief systems of the world into a select few later into the generation, and how it got there (like watching a Civ4+ game reach religious victories) and challenges along the lines of "Pilgrims in an unholy land" type, where you have to build and protect a grand shrine fortress in the center of a dangerous cult/religious area. Just building a tiny shrine out in the open is enough to enrage the surrounding settlements to ravage you worse than a gobbo invasion. Of course, trades could pass news across the lands about the shrines and etc., since they do visit the fort, much of the time, first. Pilgrims visiting could also add to the risk factors, especially if their trek crosses rival holy lands and such. Rivals find, maul and etc. the poor sap, learn of your location through them, and well, add another potential invader to the list. This should also add another dimension to the Adventure Mode as well. Especially if you intend to play the pilgrim role, hire escorts to protect you as you visit places and shrines and such, and pass on knowledge/teachings of your god (or draw blood of your victims, or their first-born, if they're that kind of god as well). Books would also have additional dimensions as well, as you can pass on/deliver your religion's bibles from settlement to settlement. Combat monks/priests would also become a possibility as well. A spiritual way to combat necromancers can be added as an expansion to the magic and etc. already going on anyway (certain religions can give certain boosts and sacrifices as well (some are non-booze religions; for example, but despite a lack of booze, truly devote followers get a boost to their endurance and focus, or something; drinking the blood, poison, or even bile of something/someone can also have effects as well, due to the sacrifice/perk effects as well, according to the generated religion(s) of the land). You can always mod it out (being RAWs), and play Full-Atheist world as well (it's more a neutralizing of the religion's effects, rather than completely eliminating them (no, vegetarian religions don't make you burst into flames if you eat meat anymore (lucky chance for that cause/effect 'sacrifice'; only applies to zealots and ☼Followers☼ (complete devotion))). I don't know how complex this feature could get though).
Ah, the mayhem of capturing/defending a dwarven Jerusalem. Hectic enough in history; just imagine something like that in DF. You could start dealing with crusades.
A (un)holy force of Crusaders/Cultists have marched onto your land. DEFEND THE SHRINES/YOUR PILGRIMS!!!EDIT EDIT:
Sorry if that derails this thread at all. Once an idea like that comes to mind, I just keep at it, until I can't think up anymore to add to it/elaborate.