A flight which follows a constant True course (a track which makes the same angle with all meridians of longitude) is called a Rhumb Line. A Great Circle, the shortest distance between two points on the Earth's surface differs from a rhumb line, unless the Rumb Line also lies along a great circle (the Equator and the meridians of longitude are the only examples). Most Rumb Lines spiral towards the pole in a shape that has the mathematical name of "loxodrome", thus rhumb lines (excluding north/south tracks, and the equator) lie on the equator side of the great circle track, and are concave to the pole.
Who your true course will vary as you fly along a Great Circle track changes depending on which hemisphere you are flying in and which direction you are travelling.