In this contribution, we propose an isogeometric collocation (IGA-C) \cite{Auricchio} formulation for the explicit dynamics of geometrically exact Timoshenko beams featuring high-order time accuracy. Past efforts addressing the explicit dynamic problem of shear-deformable geometrically exact beams were devoted to extend $\SO3$-consistent time integrators for rigid body dynamics to the beam case \cite{Marino1,Marino2}. Although this extension represented a significant advancement, time accuracy was restricted to second order, and prevented to fully exploit the well-established high-order accuracy in space of IGA-C due to the dominance of the temporal error. While this drawback has been already overcome for bi-dimensional mechanical problems using a combination of explicit Runge-Kutta time-integrators with a predictor multi-corrector algorithm \cite{Evans}, it is still an open problem for the dynamics of mechanical systems undergoing large rotations, as is the case of Timoshenko beams. We tackle this issue proposing a fourth-order time-accurate method based on the Runge-Kutta-Munthe-Kaas time integrator \cite{RKMK} for the solution of differential problems evolving on manifolds. The classical Runge-Kutta scheme is reformulated to solve the nonlinear governing equations evolving on the beam configuration manifold, $\mathbb{R}^3\times\SO3$, discretized in time and space. Numerical applications, covering different combinations of boundary conditions, will show the ability of the proposed approach to achieve high–order time and space accuracy.