Zone O is like a game of chess- you spend a while moving your pieces around the board, not making much apparent headway, trying to find the weakness in the other team's defense, and then suddenly you see your opportunity to capture their queen and it's off to the races! Zone O also often showcases the triumph of craftiness over athleticism. You don't need to be faster or taller or quicker than the other team- you just need to be skilled with the disc, disciplined, and know how to take advantage of the opportunities that present themselves.
Handlers in a zone need to be quick and have good breakmark throws- or at least have a very quick release so that they get rid of the disc before the mark can adjust. They don't necessarily need to have great hucks, since it's rare that your hucking opportunities will come from behind the cup. They do need to have great chemistry and experience working together, so they can advance the disc quickly.
Many teams have begun implementing a 2 handler zone O. The idea behind having 2 handlers is that you're more likely to have a numbers mismatch downfield (since most zone D's have at least 3 people in the cup). With 2 handlers behind the cup, your options are to either swing the disc around to a wing (where the wing is acting like a more traditional 3rd handler), or try to advance the disc through the cup. One way of getting through the cup is to have a handler "crash" the cup (run up into the cup and catch the disc inside the cup, preferably right in front of one of the other cup members). If the handler catches the disc in this position, it is easier to throw through the cup since the shortened distance between the thrower and the cup means there's less time and range for the cup to be able to intercept those throws.
Once your handlers are able to reliably get the disc through the cup, the next step is to make sure they have open targets to hit in the small window of time they might have to get their throw off. The key to this is the numbers mismatch. If you try to make sure that you arrange your upfield players in such a way as to have a numbers mismatch in the "layer" of players directly behind the cup, you will increase your chances of being able to capitalize on that numbers mismatch (for example, by having 2 poppers being guarded by 1 short deep).
Now if you have handlers that can get the disc through the cup and receivers that are open behind the cup, the next step is to capitalize on having broken the zone: keep the disc moving so that the cup can't re-form around you (give-and-go's, quick short passes), and look for long strikes. This is where your huckers come in- if you can afford to put a long thrower in the popper position, they'll be in a much better position to get a long throw off since they won't have a cup around them. Typically you'll find that the deep strikes are more effective from the wings than from the deep position, since the deep is often already being guarded by the defensive deep-deep. The offensive deep is actually in a great position to cut in for a popper, since the defensive deep-deep is unlikely to follow them in (and if they do, well, there go the wings...!).
For a zone offense to be effective, the disc must be kept moving. Any stagnation allows the defense to set up, take stock of where the threats are, and regain the advantage. If you keep the disc moving, even if it's just with short easy throws, you will prevent the defense from being able to adapt quickly enough and will eventually find the holes in their formation.