Nanomechanical resonators with fundamental mode resonance frequencies in the very-high frequency (VHF), ultra-high frequency (UHF) and microwave L-band ranges are fabricated from monocystalline silicon carbide (SiC) thin film material, and measured by magnetomotive transduction, combined with a balanced-bridge readout circuit. For resonators made from the same film, we measured the frequency dependence (thus geometry dependence) of the quality factor. We have seen a steady decrease of quality factor as the frequency goes up. This indicates the importance of clamping losses in this regime. To study this source of dissipation, a free-free beam SiC nanomechanical resonator has been co-fabricated on the same chip with a doubly clamped beam resonator operating at similar frequencies. Device testing has been performed to directly compare their characteristics and performance. It is observed that a significant improvement in quality factor is attained from the free-free beam design. In addition, from studies of resonators made from different chips with varying surface roughness, we found a strong correlation between surface roughness of the SiC thin film material and the quality factor of the resonators made from it. Furthermore, we experimentally studied the eddy current damping effect in the context of magnetomotive transduction. A high-aspect ratio SiC nanowire resonator is fabricated and tested for this study. Understanding the dissipation mechanisms, and thus improving the quality factor of these resonators, is important for implementing applications promised by these devices.