Merhaba.

On a website I´ve seen pictures of a group of Turkish people doing a kind of demonstration. They held a banner with the following statement: "Bu millet her daim dogrudan yana!".

Can you help me translate this? I can figure out only "This nation is always from the truth to the side"? But it doesn´t make much sense to me =(

It means this nation is always on the side of truth. 

Hmm.. and why don´t you say Bu millet her daim doğrunun yanında?

Is it correct too? Or only the original form is correct?

Yeah it is correct. But we also have "yana olmak".

http://tureng.com/search/yana%20olmak

"Bu millet her daim doğrudan yana!"

her zaman, her daim* (always)  [*çok nadir kullanılır (it is used too rarely)]

yan (nearside, beside)

doğru+dan(ablative)---------->doğrudan (from the truth) [otherwise, "doğrudan" means "directly" indeed, but it is not in that meaning here.]

yan+a(dative)---------------->yana (to the nearside, to the side)

doğrudan yana (may be translated as: inclined to the truth, to the side of truth, tend to truth, ...)

Bu millet her daim doğrudan yana. (This nation always supports the truth. (direct: This nation always tends to the side of truth.))