What do you say?
Let's say, I am an American citizen. But, my grandfather is a Turk.

Which one should I say?
I am a Turkish heritage.
I am a Turkish descent.

Or, what do you say?

Question

What do you say?
Let's say, I am an American citizen. But, my grandfather is a Turk.

Which one should I say?
I am a Turkish heritage.
I am a Turkish descent.

Or, what do you say?

Answer

I am of Turkish Descent. 
I have Turkish Heritage.
If it is a very recent relative, like your parent or grandparent, it is also very common to say
"My mother is Turkish." or "I am half Turkish."  or "I am one quarter Turkish." "My grandfather is Turkish."

Answer

I am of Turkish descent

I am of Turkish heritage ... or ... I have Turkish heritage

Answer

I am of Turkish DESCENT

Answer

I am of Turkish descent.  I have Turkish heritage.  I am of Turkish heritage.  Any of these are good.