Precise thermal neutron capture γ-ray cross sections σγ for 39,40,41 K were measured on a natural potassium target with the guided neutron beam at the Budapest Reactor. The cross sections were internally standardized using a stoichiometric KCl target with well known 35 Cl(n,γ) γ-ray cross sections [1,2]. These data were combined with γ-ray intensities from von Egidy et al [3] and Krusche et al [4,5] to generate nearly complete capture γ-ray level schemes. Total radiative neutron cross sections were deduced from the total γ-ray cross section feeding the ground state, σ0 = Σσγ (GS) after correction for unobserved statistical γ-ray feeding from levels near the neutron capture energy. The corrections were performed with Monte Carlo simulations of the potassium thermal neutron capture decay schemes using the computer code DICEBOX where the simulated populations of low-lying levels are normalized to the measured cross section depopulating those levels. Comparisons of the simulated and experimental level feeding intensities have led to proposed new spins and parities for selected levels in the potassium isotopes where direct reactions are not a significant contribution. We determined the total radiative neutron cross sections σ0( 39 K)=2.28±0.04 b, σ0( 40 K)=90±7 b, and σ0( 41 K)=1.62±0.03 b from the prompt γ-ray data and the γ-ray transition probability Pγ (1524.66)=0.164(4) in the β − decay of 42 K in a low-background counting experiment.