We report an experimental demonstration of room-temperature spin transport in n-type Ge epilayers grown on a Si(001) substrate. By utilizing spin pumping under ferromagnetic resonance, which inherently endows a spin battery function for semiconductors connected with a ferromagnet, a pure spin current is generated in the n-Ge at room temperature. The pure spin current is detected by using the inverse spin-Hall effect of either a Pt or Pd electrode on n-Ge. From a theoretical model that includes a geometrical contribution, the spin diffusion length in n-Ge at room temperature is estimated to be 660 nm. Moreover, the spin relaxation time decreases with increasing temperature, in agreement with a recently proposed theory of donor-driven spin relaxation in multivalley semiconductors.