That isn't how it works. The genes which code for "appearance" if that's what you mean by "traits" are in your autosomal dna; they aren't carried by your mtDna or yDna. Whatever you inherit from your mother in terms of autosomal inheritance comes from all of her ancestors, male and female, and the same is true for your father. Your mtDna and yDna ancestors are only two lines out of many.
Let's take a simple example. Let's say that a person has maternal ancestry from the Cape Verde Islands. I don't know if you're familiar with them, but the people there have quite a bit of SSA admixture. It shows up in autosomal tests, and also shows up in "appearance". I once went to a Cape Verdean festival in Massachusetts, and you can clearly see it in these people. However, the mtDna of some of them might be a totally unremarkable mtDna "H" subclade common in Portugal. (Of course, it could also be mtDna L3) The SSA could have entered through the paternal ancestors of the mother, or could indeed have entered through a maternal line, but not the one represented by the mtDna. Do you see how it can work?
If you follow this link you can access a lot of "beginner's guides" to the whole subject of genetic genealogy. You might find them useful.
http://isogg.org/wiki/Beginners'_guides_to_genetic_genealogy
I take it you're not an afficionado of cheese?
Have you ever heard of Roquefort, Manchego, or, if you're in a goat cheese mood, feta, chèvre (Bucheron), garrotxa...well, I could go on and on. Spain is actually the largest producer of sheep cheeses. Are you going to attribute it to J1 sheep herders? Bucheron is made in the Loire Valley, feta in, obviously, Greece. I'm not aware of any "Arabic" domination of those areas.
The goat, the sheep, and the cow were all domesticated in the Near East and were all brought to Europe by Neolithic farmers.
http://anthropology.si.edu/archaeobio/images/r4_468.png
This is a paper on domesticated animals in the Early Neolithic in the Balkans:
http://www.academia.edu/4124374/Ani...y_Neolithic_of_the_Balkans_and_Central_Europe
Animal Herding in the Balkan Neolithic:
http://www.academia.edu/1517144/Herding_settlement_and_chronology_in_the_Balkan_Neolithic
Although there are nutritional pros and cons to the milk of all these animals, in one thing the cow has the others beat hands down, and that's in the total quantity of milk they can produce, which makes sense given the size differential. However, the downside is that they require a great deal of food, huge quantities of grass or hay or even better, grain. In certain parts of Europe that's just not to be had. Terrain and climate dictate what you grow and which animals you domesticate, the "terroir", or at least they did. That's why in rural Emilia there are more cows than people, but in Liguria and the Lunigiana people could only manage a cow or two for personal use, and sometimes not even that, and the vast majority of cheeses are made from either sheep's milk or goat's milk. Oh, and we don't have much J1 at all.
It's easy, in the beginning, without an understanding of the causes of certain economic paradigms or cultural manifestations, to draw false conclusions. For example, I recently read a post elsewhere where someone, after listening to some "Ashkenazi' music and some "Sicilian" music, saw some similarities between them and tried to use that as "proof" of gene sharing between them. This totally ignores the fact, as others pointed out, that cultural flow, particularly in the modern era, can have absolutely nothing to do with gene flow. More directly, if this person knew anything about European music, he would have known that the two pieces of music were both influenced by music forms created in eastern Europe. Do you know how much Italian "folk" music is set to mazurkas and polkas? Why, the older folks wouldn't be able to dance at summer festivals if all such music was removed from the program.
*The same thing happened with the "jig". It was created or evolved in 16th century England. From there it spread to Ireland, France, Italy etc. (the French gigue, the Italian giga) This isn't proof of any gene flow or sharing between them.
https://en.wikipedia.org/wiki/Jig
What makes you think we don't recognize that this is all very complicated? See:
http://www.eupedia.com/forum/thread...the-Fertile-Crescent?highlight=Post+Neolithic
As for the yDna of the Neolithic, you might want to read Iain Mathiesen et al, which shows that the Anatolian farmers carried various yDna clades, even if the majority were G2a. That doesn't mean there was only one Neolithic wave, necessarily. J2 and the immediate precursor to E-V13 suddenly show up in Europe in the mid-Neolithic.
http://biorxiv.org/content/biorxiv/early/2015/03/13/016477.full.pdf
*Ah, I forgot all the songs based on the German waltz. It's all called ballo liscio in Italian, if anyone is interested, dances (and songs) based on waltzes, mazurkas and polkas. Later on, tangos were added, and there was no mass migration of Argentinians to account for it.