I never understood what "diye" means . If somebody can explain me please ? 

+ 1 - I can´t get my head round this either.

 Gelmedin diye çok üzüldüm.

I was sad as you didn´t come.

Gelmiş diye duydum.

I heard that (meaning - from someone as) he came.

Ağlama diye söylüyorum.

I am telling you so that you don´t cry.

As a result you use it in many ways just like şey, but there is no rule for it.

Doesnt come from "demek" ?

 Yes, it does.. but it doesn´t exactly mean "demek" when it is used in this way.

it can also mean "in case"...

I´ve seen ´diye´ used before other verbs in a children´s story I read - like this:

´Ne var bunun içinde? Demir mi?´ diye sordu babam´

and

´Susun!´ diye bağırdı trafik polisi

....which I read as

´What´s inside this? Iron? asked my father

and

´Quiet! shouted the traffic policeman

Therefore, in this case, is diye used to emphasıse the verbs sormak and bağırmak?

That´s right but this is not an alternative.

So;

´Ne var bunun içinde? Demir mi?´ diye sordu babam´ - correct

´Ne var bunun içinde? Demir mi?´ sordu babam´ - wrong

´Susun!´ diye bağırdı trafik polisi - correct

´Susun!´  bağırdı trafik polisi - wrong

But ´dedi´ is an exception.

´Ben gelmeyeceğim´ dedi. - correct

´Ben gelmeyeceğim´ diye dedi - wrong

Is it used on spoken language or more often in written language ?

it is used on spoken and written langugae, but mostly preferred in spoken language.

One more example:

Ne diye bağırıyorsun bana şimdi? why are you yelling at me now?

Ne diye? =Neden?

We can use ¨diye¨with different meanings

PS: Sorry about my English