Which NFL team has traditionally played on Thanksgiving since 1934?

It is the Detroit Lions.

Since 1934, the Lions have played on every Thanksgiving Day (except for 1939 and 1940, when the Pittsburgh Steelers and the Philadelphia Eagles played), and 1941-1944 (when no game was played on Thanksgiving Day).

As of the 2008 season, the Lions' record on Thanksgiving Day is 35-32-1.