The good news is that Azerothian zombies, just like in most movies, move slow. Just be sure to stay farther than 21 yards away from them as they have Retch! a ranged spell that will slow anyone within the area of effect. This is the smoky green cloud on the ground you see in all those screenshots. They can also summon a burst of speed using Lurch! so don't be too confident that they won't close the distance. That ability also frees them from movement impairing effects so trapping them or slowing them isn't a guarantee you'll be able to stay away from them. The idea is to kite the zombies, because getting into melee range with them is a very bad idea.
Keep your distance
A zombie's staple attack is Mangle!, not to be confused with the exclamation point-less Druid spell of the same name. The attack itself isn't anything you should worry about, it's the risk of infection as every melee attack has a chance of infecting you with the plague. As a melee class, you're better off using your ranged weapon to strike them from afar. Otherwise, you should have a pocket healer that can remove the disease from you. Although reports have been pouring in that the disease is becoming harder and harder to cure...
Murder by numbers
If you watch a lot of horror movies, you probably already know that it's unwise to take on a lot of zombies on your own. Most of their abilities confer their effects to other zombies, so they are much more powerful with company. Smart groups of zombies will take turns using their cooldowns in order to get the best effect. This means that a party of zombies can pretty much keep most of their buffs up indefinitely with ordered usage. Even if you get help from city guards, these NPCs will get infected and turn. In fact, the more zombies there are, the faster they -- and you -- will convert because zombie auto-hits shave ten seconds off the disease's duration. This means six hits will shave one minute off, and a lot of hits from a lot of zombies can mean a very grim end.
Make sure to keep on curing the disease if you have the ability to do so. Even though 10 minutes seems like a long time, the melee hits I mentioned earlier quickly cuts that time down and you can suddenly turn into a zombie before you're aware of it. You can also always retreat to an Argent Healer, who will cure your affliction within close proximity. Also be aware that using immunity mechanics such as Divine Shield or Ice Block while infected will transform you into a zombie immediately, rather than removing the disease from you.
You're fighting Undead
Remember that zombies are classified as Undead, which means abilities such as Holy Wrath, Turn Evil, Shackle Undead, and Exorcism all work on them. This is a great opportunity for Paladins and Priests to use abilities normally reserved for NPCs on fellow players. Holy Wrath also works as an AoE stun, so it's a good ability to use in the midst of the zombie masses.
Remember that zombies can be of any level -- from lowbie players giving townsfolk a scare to Level 70s wreaking havoc in major cities. Don't hesitate to strike down low-level zombies. They have exactly the same abilities as the high level ones, so they're just as dangerous except that they are easier to kill because they have less life. If that weapon swapping bug is fixed, low level zombies should go down fairly quickly. If you kill them, they will return uninfected and hopefully stay that way.
How has your crusade against Arthas' minions been coming along? Are there zombie fighting tips and tricks you'd like to share? Let us know!