Large eddy simulations of vertical convective vortices and dust devils in the Martian convective boundary layer are presented, employing a version of the Mars MM5 mesoscale model, adapted to use periodic boundary conditions and run at resolutions of 10 to 100 m. The effects of background horizontal wind speed and shear on dust devil development are studied in four simulations, each extending over the daytime portion of one Martian day. The general vorticity development in all cases is similar, with roughly equal positive and negative vorticity extrema. Two dust devils were found to develop in the highest wind speed case and in a case run without background wind. The dust devil structures were found to agree well qualitatively with terrestrial dust devil observations, including the prediction of greatly diminished vertical velocities in the vortex core. Thermodynamic scaling theory of dust devils was found to provide good prediction of the relationship between central pressure and temperature in the modeled vortices. Examination of the turbulent kinetic energy budgets suggests balance between buoyancy generation and loss through dissipation and transport. The vorticity for the dust devils is provided by twisting of horizontal vorticity into the vertical. The horizontal vorticity originates from horizontal variations in temperature at the lower boundary (thermal buoyancy). While the horizontal winds generated by the modeled dust devils were likely insufficient to lift dust, this study provides a solid starting point for dynamic modeling of what may be an important component of the Martian dust cycle.