The problem is actually a critical one. Externally, humans are pretty symmetrical. Internally, things are much more complicated. Your left-lung has two lobes, your right has three. Your heart nestles gently in between the middle, but slightly towards the left (which is why the left side is missing a lobe – it’s to fit the heart). Your liver sits to the right, while the other organs are packed around in a non-symmetric way.
There is a group of the population which have this all confused.
Situs invertus is a disorder affecting some 1:8500 people, and which results in (internally) left-being-right and vice versa. Their heart sits to the right, and their right lung has two lobes; their left lung has three. Then there are the unfortunate people who are born with no left-right dissymmetry at all. Swapping left and right seems like an easy thing, when you think about it. Everything would pack normally. If the embryo doesn’t know where to put things, however, then vasculature tends to just connect to whatever is closest – and isomerism results in babies having emergency surgery within hours of birth in order to re-plumb pretty much everything inside.
So how do you figure all this out? Top-bottom and front-back are all formed by the interactions involving Hox genes. Basically, a cluster of cells known as the ‘node’ produces some chemicals which act as transcription factors, and then, as the chemicals get diluted, form a gradient which enables a ‘top’ and a ‘bottom’.
Left and right is much cooler.
It turns out that sitting on the node is a clump of cilia swirling in a counter-clockwise direction. Nobutaka Hirokawa at the University of Tokyo showed that these cilia produce a leftward flow of fluid over the embryo which is therefore able to give you a left-to-right gradient. The node produces another gradient of factors which flows left to right, giving you a difference in sides. If you culture some mouse embryos, stick a stirrer inside and artificially make the flow go the opposite direction, as Hiroshi Hamada from Osaka University did, you end up with mice suffering from situs invertus.
So, regardless if you're left or right handed... be glad that you can tell left from right.