We propose a practically stable explicit finite difference method (FDM) for the ternary Cahn–Hilliard (CH) system, which involves challenging fourth-order nonlinear terms. Therefore, if we use a fully explicit numerical method such as the explicit Euler scheme, then the excessively restrictive time-step limitation makes it impractical to use. The proposed algorithm is based on the alternating direction explicit (ADE) scheme, which ensures both simplicity of implementation and stability of the computational solutions for the ternary CH system. To show the high performance of the proposed algorithm, we present multiple numerical experiments. The numerical tests demonstrate that the proposed method overcomes the stringent time-step limitations inherent in the explicit Euler method.