I too am searching for distant members of the Fatseas family. My name is Stephen (Steve) Fatseas. I was born in New York City in 1945. My grandfather's name was Eugene (may be Americanized) Fatseas and my grandmother was Maria Fatseas (maiden name possible Lamar).
My father James (Dimitrios) emigrated to the US in the early 1900's. The Ellis Island archives lists him arriving on the Themistocles on July 9, 1914 at the age of 8 along with his older brother Nicolaos, age 10. Both have their birthplace listed as Smyrna, Turkey. I can't find info on my grandfather's entry on the same ship, but it was common for heads of households to come over early and send for their families later.

The 1930 US Census lists my grandparents and the family as living in Wayne County, Michigan (probably Detroit).
In the 1930 Census:
Maria, age 50
Eugene, age 50
Nick (Nicolaos), age 27
Jimmey (James/Dimitrios), age 24
Hellen, age 14 (born in US)
Daisie (Anastasia), age 13 (born in US)

Both my Ya-Ya and Papou were born circa 1880, but I don't have the information as to where. I also don't know when they arrived in Smyrna, but I do know they lived there for a few years before coming to America. My Papou died circa 1930 and my Ya-Ya died in 1964. My father died in 1966.

My mother and father had 4 sons, Eugene of Topeka, KS, USA, James of Corona, CA, USA, Frank of Olympia, WA, USA, and myself.

Any further information would be appreciated. You may contact me at steve.fatseas@intermec.com