The dynamic behavior of rotating beams, including rotor blades, spinning space structures and gas turbines is of great practical interest. A recent computation approach called the Adomian modified decomposition method (AMDM) is used to carry out the free transverse vibration analysis of rotating non-uniform Euler-Bernoulli beams using several boundary conditions, rotating speeds and beam lengths. The AMDM allows the governing differential equation to become a recursive algebraic equation and the boundary conditions become simple algebraic frequency equations suitable for symbolic computation. With additional simple mathematical operations on the model, the natural frequencies and corresponding closed-form series solution of the mode shape can be obtained simultaneously. For verification of the AMDM method, numerical examples are analyzed with boundary conditions imposed, including clamped-free, with attention given to convergence of the solutions. As the AMDM technique is systematic, it is found straight-forward to modify boundary conditions from one case to the next. Comparison of results is given with published data and it is found that the current results are in close agreement with those in the literature.