Of course, Thanksgiving as we know it today did not originate with the Pilgrims as many suppose, though their feast is the model for it. This is one of the myths circulated about Thanksgiving that even most Americans were taught in school. 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, days of Thanksgiving in early America called for a national day of prayer and fasting, due to a lack of ecclesiastical feasts by Protestants, 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. In 1863 President Abraham Lincoln called on all Americans to observe on the last Thursday of November a day of Thanksgiving and Praise for all the good things God has bestowed on us as individuals and as a country, and since that time every American President has followed his example to make a similar proclamation. For those who follow the New Calendar, 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 pastoral dispensation, which was initially granted by the Ecumenical Patriarchate, is given by local bishops and parish 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 individually, as a family and as a country. The Nativity season is usually a fairly lenient fasting time as well, especially before mid-December, so such "economia" should not be looked upon as a major violation of any canonical rule (see here). Also, as Christians, the Apostle Paul advises us to not give offense either by our fasting or our feasting, and certainly it would be odd for Orthodox Christian citizens in America to not join the rest of the country in commonly celebrating a Thanksgiving feast to the glory of God, whether it be with the traditional meats or not, after all, even many atheists celebrate some sort of Thanks on this day as well. "Therefore, whether you eat or drink, or whatever you do, do all to the glory of God" (1 Cor. 10:31). Ideally it is recommended for Orthodox to participate in the Divine Liturgy on Thanksgiving morning prior to feasting, since the Eucharist is the ultimate offering of thanksgiving to God. An Akathist of Thanksgiving is also available.