Images in 3.5-30 keV X-rays obtained during the first few minutes of seven solar flares show rapid motions. In each case X-ray emission first appeared at one end of a magnetic field structure, and then propagated along the field at a velocity between 800 and 1700 km/s. The observed X-ray structures were 45,000-230,000 km long. Simultaneous H-alpha images were available in three cases; they showed brightenings when the fast-moving fronts arrived at the chromosphere. The fast-moving fronts are interpreted as electron thermal conduction fronts since their velocities are consistent with conduction at the observed temperatures of 1-3 x 10 to the 7th K. The inferred conductive heat flux of up to 10-billion ergs/s sq cm accounts for most of the energy released in the flares, implying that the flares were primarily thermal phenomena.