Residual stress occurs in the materials after different methods of processing due to the application of pressure and/or thermal gradient. The occurrence of residual stresses can be observed in both subtractive and additive-manufactured (AM) materials and objects. However, pressure residual stresses are considered, in some cases, to have a positive effect; there are applications where the neutral stress state is required. As there is a lack of standards describing the heat treatment of AM materials, there is a need for experimental research in this field. The objective of this article is to determine the heat treatment thermal regime to achieve close to zero stress state in the subsurface layer of additively manufactured AM316L stainless steel. The presented objective leads to the long-term goal of neutral etalons for eddy current residual stress testing preparation. A semi-product intended for the experiment was prepared using the Selective Laser Melting (SLM) process and subsequently cut, using Abrasive Water Jet (AWJ) technology, into experimental specimens, which were consequently heat-treated in combination with four temperatures and three holding times. Residual stresses were measured using X-ray diffraction (XRD), and microstructure variations were observed and examined. A combination of higher temperature and longer duration of heat treatment caused more significant stress relaxation, and the original stress state of the material influenced a degree of this relaxation. The microstructure formed of cellular grains changed slightly in the form of grain growth with randomly occurring unmolten powder particles, porosity, and inclusion precipitation.