When starting up, you have no weapon and no mean to defeat zombies at all. Are there ways to lose a zombie aggro or multiple zombies, even?
Run some more!
Lather, rinse, repeat. Eventually, zombies will stop chasing you. According to the DayZ wiki:
Breaking line of sight is paramount. Sometimes simply going prone can throw a zombie off you scent. Another important thing to remember is zombies cannot run in buildings. Taking a route through a building with multiple exits will allow you to slow down your pursuer significantly. If you have the ammo and don't mind attracting the attention, this will also allow you to easily take them out with a well placed shot to the head.
Other alternatives include swimming, climbing, and closing doors. Although zombies can do all these things as well, it can slow them down.
There are a few key things to know about Zombies when trying to run away from them (from the DayZ wiki)
Given this, the methods I find best are
Run through buildings
Zombies can't run through buildings, so a good tactic is to either run through a building, or run into a building, wait for the Zombies to move away from the entrance and then peg it out the way you came in.
Make sure that you don't try this in small buildings with only one entrance as you will probably just back yourself into a corner. Also be careful - zombies can attack (and sometimes even walk through) walls and closed doors.
Run up or down hills
If you manage to find a hill steep enough so that you slow to a walk going up it then try zig-zagging up the hill - the Zombie too will slow to a walk and running up hills in a zig-zag is much faster that trying to walk directly up the hill, so using this method is a quick way to loose your Zombie if you have any suitable hills nearby.
You can also give this a try on steep downards hills - last time I tried this the Zombies comically threw themselves off the hill to their deaths.
Run through bushes
Bushes and some trees can break line of sight - running through a dense line of trees or bushes can in itself be enough to loose a Zombie if there are enough bushes.
Run around buildings
If you don't have any suitable buildings / bushes / hills nearby then if you are desparate you can just try running around a building / fences. Remember that the goal is to break line of sight, and then get far enough away so that the Zombie doesn't find you again. This method is less than reliable, but can sometimes work if you are lucky.
In all cases as soon as you see that the Zombie has stopped running after you switch yourself to a crouch walk (so they don't just immediately spot you again / to stop you attracting even more zombies) and get away as fast as you can.
Best thing to do is abort if you aggro zombies, they all disappear when you log back in.
In my opinion the best way is to get inside buildings with 2 entrances, like large barns. You get inside, wait until all of the zeds are inside walking slowly and leave by the other entrance. If you want to make really sure you lost them, after leaving go prone and take some distance. Almost never fails, unless some zombie decides to go in by the second entrance.
Once you break Line of sight zombies will stop chasing you. A common looting tactic is to just sprint into town take what you need and then wind around some buildings and run away. Right now this works because they took out the fatigue system. I don't know if there are plans to put it back in.
Also be aware that due to a bug, zombies glitch through closed doors.