Three-dimensional thermocapillary convection of a Boussinesq fluid in an open cubic container is considered in the absence of buoyancy. The steady motion is computed with a modified Full-Approximation Scheme, Full Multi-Grid (FAS-FMG) method based on a finite difference, primitive variable formulation. Solutions on a staggered grid of 32 x 32 x 32 points for each unknown are obtained by a line coupled, block implicit relaxation method for the Navier-Stokes equation, and an alternating zebra-relaxation method for the energy equation. Three-dimensional (3D) flow structures and temperature fields are presented for Reynolds numbers up to 5 x 10(4) and Prandtl numbers Pr = 0.01, 1, 10 and 50. Apart From the main vortex known from 2D calculations pronounced secondary vortices appear. The mechanisms driving this secondary how are discussed in terms of well-known properties of the main vortex.