A lattice Boltzmann model for the dissolution of solid structures of arbitrary shape in multi-component liquids is developed. To model diffusion-controlled dissolution, a multicomponent boundary condition is presented to impose a fixed concentration on an arbitrarily located boundary. The dissolution rate of the solid is calculated based on the diffusion flow in the boundary layer. The model is validated using analytical solutions of simple dissolution problems in a static fluid, and is applied to the dissolution of a cylinder in a laminar flow.