The inadequacy of traditional theory of elasticity in describing such a phenomenon as dispersion associated to a propagating wave with wavelength comparable to the intrinsic length of the medium of interest is well-known. Moreover, under certain circumstances it is incapable of capturing all the propagating waves. A remedy to such dilemmas is the employment of the more accurate higher order continuum theories which give rise to the appearance of at least one new characteristic length in the formulation. The experimental evidences as well as lattice dynamic analysis suggest that, although higher order continuum theories result in some improvements, but cannot fully overcome the above-mentioned dilemmas, unless the micro inertia term is included in the formulations. The current work addresses the elastodynamic fields of an anti-plane shear wave scattered by a micro-/nano-fiber embedded in an infinite matrix using couple stress theory with micro inertia term. Moreover, the formulations pertinent to the cases where the incident wave strikes an embedded micro-/nano-size circular cavity or a rigid immovable micro-/nano-fiber are also obtained. Within this theory, the appearance of a new length scale, so-called "dynamic characteristic length" stems from consideration of the micro inertia, which gives rise to physically realistic dispersion relations with characteristic resembling those observed in experiments. The effects of two different types of boundary conditions for the cases of elastic and rigid immovable fiber encountered within the present theory are discussed. By using this theory, the corresponding analytical expressions of the elastodynamic fields, total and differential scattering cross-sections, and the dynamic stress concentration factor are presented and their dependence on the characteristic lengths and frequency are examined. It has been shown that the effect of micro inertia term is more noticeable in higher frequencies.