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By virtue of the strong confinement of surface-wave fields that it provides, the transverse corrugated conducting rod is an important structure in the fields of plasmonics and metasurfaces, finding applications in focusing, sensing, imaging, spectroscopy, and subwavelength optics, among others. This paper presents an analytical modal method for treating such grated rods with generally dielectric-filled grooves, one which offers rapid yet accurate surface-wave modal solutions. Based on the asymptotic corrugation boundary conditions, the formulation is simple and elegant, providing not only the dispersion relationship between the frequency and wavenumber but also the explicit functional forms of the fields. Dispersion and modal field results obtained by the proposed method are validated with an independent full-wave solver. Because of its candidacy for microwave applications at high powers and high frequencies, as well as transmissions over long distances, studies of dielectric and conductor losses are also carried out, for both the grooved rod and its likewise corrugated circular waveguide counterpart. Parametric studies are conducted on three aspects, namely, the degree of field localization on the surface of the rod, as well as attenuation due to dielectric and metal losses. Measurements of dispersion and field decay properties conducted on a manufactured rod that concur with theory are also reported.