I'm rather late to join in with answers, but this looks like a good reference question and no one has yet mentioned our fool proof method.
There's two issues that need to be addressed to have tangle free rope, wire, cable, etc. One is you must not introduce twists in the material, as kinks will develop. Some variation mentioned already of alternating direction or making figure 8s as it's coiled will prevent twists.
The other issue is the loops getting interlaced as the coil is moved and jostled. The best defense is to keep the loops short and tidy and avoid jostling them. That is often impractical. In high angle rescue, tangled ropes can make the difference between life and death. We also store our ropes in bags to protect them from dirt, grease, etc. We also like open coils we can throw over our heads and easily carry on one shoulder and across the chest. 300' of 1/2" static line can get really heavy in a short distance.
There's only one method we've found that will reliably uncoil without twists or tangles after being tossed around in bags and carried for miles on various shoulders. We call it a bird's nest. It's a circular variation of a daisy chain. It takes a while to do it when breaking down after a rescue, but it's worth it in time saved on the next rescue, and it's not that bad when two people work together, though one person can do it well enough.
You start with three simple, good sized loops of rope at one end. Then you start by tying a half hitch on a bight around the three loops, loosely binding them together.
Then in classic daisy chain technique, a bight of rope is passed through the previous loop, except in this case, each bight also wraps around the coil, such that, as the daisy chain progresses, the initial 3 loops will be fully surrounded by interlocking loops.
Once the daisy chain goes all the way around, it continues by wrapping the first layer of loops with another layer of loops.
Once the layers of loops get fairly fat, the daisy chain work starts going pretty fast, as each loop requires a decent amount of rope to reach around all the material. When the rope's free end is reached, it's finished by pulling the free end through the final loop.
For a little extra security, we cinch the free end around the coil with a simple overhand knot (this last bit is not shown in the above image). It looks like a mess, hence the name "Bird's Nest", but it is actually an ordered series of untwisted loops that are impossible to disrupt into a tangle during handling.
When bringing the rope into service, simply undo the overhand and pull the rope through the first loop and the remaining loops will feed out without kinks or tangles. Once one end is secured, we can reliably drop the nested coil over the edge of a cliff and nearly all of it will feed out cleanly without fail. Any daisy chain remaining at the bottom end is quickly shaken out by a rescuer when he/she approaches the end.