In addition to the traditional hollow circular sections used in marine structures, other hollow sections have attracted the attention of architects and design engineers due to their mechanical characteristics such as torsional rigidity and local strength against impact loading. The purpose of this study is to investigate dynamic response of pipes conveying fluid with variable wall thickness through both circumferential and axial directions. Pipes with variable wall thickness have different flexural rigidities about two different principal axes. This property allows these pipes to be oriented efficiently, meet various design requirements and resist the applied loads. The results of this investigation provide a better insight into the physics and dynamic behavior of non-circular pipes conveying fluid. Two different geometries are studied, (i) the pipe is assumed with a general non-circular cross section with variable wall thickness along the circumferential direction, (ii) both inner and outer boundaries of the pipe cross section are assumed to be circles whereas the wall thickness of the pipe along axial direction is varied with a specified function. The governing differential equations of the problem are derived using Timoshenko beam theory with the effect of shear deformation included in the formulation. The discretization of the problem domain is done using the finite element method. Consequently, a modal analysis is employed to calculate the critical flow velocities of the pipe with clamped-clamped end conditions. The effects of different cross sections on the critical flow velocity are investigated. The importance of Coriolis forces on the presence of coupled-mode flutter and re-stabilization point are also discussed for different values of mass ratio.