We study the approximation complexity of certain kinetic variants of the Traveling Salesman Problem where we consider instances where points move with fixed constant velocity in a fixed direction. We prove the following results: 1. If the points all move with the same velocity and in the same direction, then there is a PTAS for the Kinetic TSP. 2. The Kinetic TSP cannot be approximated better than by a factor of two by a polynomial time algorithm unless P=NP, even if there are only two moving points in the instance. \item The Kinetic TSP cannot be approximated better than by a factor of $2^{\Omega(\sqrt{n})}$ by a polynomial time algorithm unless P=NP, where $n$ is the size of the input instance, even if the maximum velocity is bounded.