We study three monodomain (single-crystal) nematic elastomer materials, all side-chain siloxane polymers with the same mesogenic groups but with different types of cross linking: (i) short flexible siloxane linkage affine to the network backbone, (ii) short flexible aliphatic cross links miscible with mesogenic side-chain groups, and (iii) long segments of main-chain nematic polymer. The dynamic mechanical response of these three systems shows a characteristically universal decrease of storage modulus and a corresponding increase of loss factor. This effect of "dynamic soft elasticity" is strongly anisotropic, depending on the nematic director orientation. We examine the important role of the average backbone chain anisotropy r(T)=l(parallel)/l(perpendicular), which is affected by the cross-linking geometry and contributes to the magnitude and frequency dependence of the dynamic anomaly, and discuss possible applications in mechanical damping and polarized acoustic technology.