Recent theoretical studies of topologically nontrivial electronic states in Kondo insulators have pointed to the importance of spin-orbit coupling (SOC) for stabilizing these states. However, systematic experimental studies that tune the SOC parameter λ SOC in Kondo insulators remain elusive. The main reason is that variations of (chemical) pressure or doping strongly influence the Kondo coupling J K and the chemical potential μ —both essential parameters determining the ground state of the material—and thus possible λ SOC tuning effects have remained unnoticed. Here, we present the successful growth of the substitution series Ce 3 Bi 4 ( Pt 1 − x Pd x ) 3 ( 0 ≤ x ≤ 1 ) of the archetypal (noncentrosymmetric) Kondo insulator Ce 3 Bi 4 Pt 3 . The Pt-Pd substitution is isostructural, isoelectronic, and isosize, and it therefore is likely to leave J K and μ essentially unchanged. By contrast, the large mass difference between the 5 d element Pt and the 4 d element Pd leads to a large difference in λ SOC , which thus is the dominating tuning parameter in the series. Surprisingly, with increasing x (decreasing λ SOC ), we observe a Kondo insulator to semimetal transition, demonstrating an unprecedented drastic influence of the SOC. The fully substituted end compound Ce 3 Bi 4 Pd 3 shows thermodynamic signatures of a recently predicted Weyl-Kondo semimetal.