Here is an excellent video recorded by a Russian native speaker teaching you how to say Happy Thanksgiving in Russian: http://www.primelanguageservices.com/blog/free-lessons/how-to-say-happy-thanksgiving-in-russian/
You will literally be saying “with the day of thanksgiving” meaning “I congratulate you with Thanksgiving”, in Russian “С ДНЁМ БЛАГОДАРЕНИЯ” pronounced as “S DNYOM BLAH-GOH-DAH-RYE-NEE-YA”.
For those of you who have been studying Russian for a while, БЛАГОДАРИТЬ / BLAH-GOH-DAH-REET’ means “to thank someone”. And the word ДЕНЬ / DYEN’ means “day”. So we are saying “with the day of thanksgiving” in Russian.
While Thanksgiving is not a Russian holiday and is not celebrated in Russia itself, many Russians residing in the United States have adopted the wonderful family tradition and get together for the Turkey Day, which brings me to the next point. Jokingly we might say to each other something like “Happy Turkey Day”, in Russian “С ДНЁМ ИНДЕЙКИ” pronounced as “S DNYOM EEN-DYEY-KEE”. Индейка / een-dyey-kah means “turkey” in Russian and is also a great word to learn for Thanksgiving.
To summarize, today we learned how to say “Happy Thanksgiving” in Russian, which is С ДНЁМ БЛАГОДАРЕНИЯ / s dnyom blah-goh-dah-rye-nee-ya. We also learned how to say “Happy Turkey Day”, which in Russian is С ДНЁМ ИНДЕЙКИ pronounced as s dnyom een-dyey-kee. We welcome any questions on “How to say Happy Thanksgiving” in Russian or anything related to Russian language and culture. Enjoy speaking Russian and Happy Thanksgiving!