In many cases, the functional performance of additively manufactured components can only be ensured by finishing the functional surfaces. Various methods are available for this purpose. This paper presents a procedure for selecting suitable processes for finishing laser beam melting additive–manufactured parts which is ultimately based on technological knowledge. It was experimentally proven that the use of several consecutive finishing processes is beneficial to achieve better surface quality. One finishing process chain was particularly effective (namely particle blasting/vibratory grinding/plasma electrolytic polishing) and the technological limits of this method were investigated in this study. The optimal parameters for this process combination ensured a surface roughness Sa < 1 µm.