Solving the eikonal equation is used widely in traveltime calculation, tomography, Kirchhoff migration etc. The complex eikonal equation governs the traveltimes in an attenuating medium, where the real and imaginary parts of the traveltimes are associated with the phase and energy-absorption, respectively. Attenuating orthorhombic anisotropy can be used to explain the azimuthal variation of velocity- and attenuation-anisotropy measured from surface seismic data. We present an approximate method to solve the acoustic eikonal equation for an attenuating orthorhombic medium. We combine perturbation theory and Shanks transform in different ways to derive the analytic solutions in the case of homogeneous media. We design a fast marching scheme to solve the acoustic eikonal equation numerically. We share some numerical examples to demonstrate the effectiveness of the complex eikonal equation in predicting attenuation.