Of course, Thanksgiving as we know it today did not originate with the Pilgrims, though their feast is the model for it. This is one of the mythologies surrounding Thanksgiving. Another myth of Thanksgiving is that it is traditional for Americans to have a feast on days proclaimed by U.S. Presidents as days of Thanksgiving. In fact, Thanksgiving in early America called for a day of prayer and fasting, not family gathering, football watching and feasting on turkey and all the other good stuff. Read more here, here and here.
Orthodox Christians in America are in a unique position regarding the celebration of Thanksgiving, which is our patriotic duty. Thanksgiving is no longer a time of fasting, but of feasting with family and friends (something which began to take shape after the Civil War in the 19th century and the Lincoln proclamation). For those who follow the New Calendar especially, the giant feast associated with Thanksgiving coincides with a period of 40-day fasting prior to Christmas, though it sometimes falls within the Old Calendar fasting period as well (when fasting begins on November 28th). Generally, however, a dispensation is granted by local bishops and priests to allow the Orthodox faithful to participate in this traditional American holiday as a harvest festival, a time for families to come together and celebrate, and to offer thanks to God for all He grants to us both individually, as a family and as a country. The Advent season is usually a lenient fasting time also, so such "economia" should not be looked upon as a big deal (see here). Ideally it is recommended for Orthodox to celebrate the Divine Liturgy on Thanksgiving morning prior to feasting, since the Eucharist is the ultimate offering of thanksgiving to God.
Below are a few helpful articles written by Orthodox Christians regarding the holiday of Thanksgiving:
Dispelling Confusion About Fasting During Thanksgiving
What is So Orthodox at Thanksgiving?
A Meal of Unity: Christian Reflections on Thanksgiving Dinner
Encyclical of Archbishop Demetrios For Thanksgiving Day 2010
The Final Divine Liturgy of Fr. Alexander Schmemann
Akathist of Thanksgiving
A Happy Thanksgiving to all!