Structural and electronic properties of acetylene and ethylene adsorbed on the Si-terminated beta-SiC-(2x1) surface are investigated by ab initio calculations based on density functional theory. A host of conceivable adsorption geometries with half-monolayer and monolayer coverages are considered. The resulting optimal adsorption configurations are quite intriguing in that they differ qualitatively from those of the related and well-studied adsorption systems of acetylene and ethylene on Si(001)-(2x1). At the beta-SiC-(2x1) surface, acetylene is found to preferentially adsorb for both coverages in a rotated bridge site forming four sigma bonds with Si substrate surface atoms. In this site, the molecule is located in the middle between four neighboring Si surface layer atoms and is oriented perpendicular to the original Si dimers of the clean surface. For ethylene adsorption, a cross dimer site is favored in which the molecule is bonded by two sigma bonds to two surface layer Si atoms of neighboring Si dimers. The ethylene monolayer forms a c (2x2) configuration. This cross dimer structure is favored by about 1.5 eV per molecule with respect to ethylene adsorption in on top positions above Si dimers. This is noteworthy, since for ethylene adsorption on the Si(001)-(2x1) surface, the on top site is most favorable. Interestingly, when ethylene adsorbs dissociatively on beta-SiC-(2x1) this leads to an even larger adsorption energy giving rise to dihydride structures with 1x1 or c(2x2) unit cells for monolayer or half-monolayer coverages, respectively. We present, as well, surface band structures and charge densities for a few adsorption configurations and discuss them in relation to the electronic properties of the clean beta-SiC-(2x1) surface and of acetylene and ethylene molecules.