In this paper, the complex viscoelastically coupled global mechanics of fluid-conveying microtubes is examined for the first time. The externally excited microtube is assumed to be embedded in a nonlinear elastic medium. A scale-dependent theoretical model is presented with consideration of curvature nonlinearity within the context of the modified version of the couple stress theory (CST). According to Hamilton's energy/work principle, the coupled nonlinear equations of fluid-conveying microscale tubes are presented. Both the transverse and longitudinal displacements and inertia are taken into account in the continuum-based model and numerical calculations. In order to discretise the governing nonlinear differential equations, Galerkin's weighted-residual procedure is employed. The bifurcation characteristics of the fluid-conveying microsystem with clamped-clamped boundary conditions are obtained within the framework of a direct time-integration procedure. It is found that the complex global dynamics of the fluid-conveying microsystem is very sensitive to the speed of the flowing fluid.