We have previously calculated the pion-nucleon ΣπN term in the chiral mixing approach with u, d flavors only, and found the lower bound ΣπN ≥ 1 + 16 3 sin 2 θ 3 2 m 0 u + m 0 d where m 0 u , m 0 d are the current quark masses, and θ is the mixing angle of the [( 1 2 , 0) ⊕ (0, 1 2 )] and the [(1, 1 2 ) ⊕ ( 1 2 , 1)] chiral multiplets. This mixing angle can be calculated as sin 2 θ = 3 8 g (0)A + g(3) A, where gA , gA , are the flavor-singlet and the isovector axial couplings. With presently accepted values of current quark masses, this leads to ΣπN ≥ 58.0 ± 4.5 +11.4 −6.5 MeV, which is in agreement with the values extracted from experiments, and substantially higher than most previous two-flavour calculations. The causes of this enhancement are: 1) the large, ( 16 3 ≃ 5.3), purely SUL(2) × SUR(2) algebraic factor; 2) the admixture of the [(1, 12 ) ⊕ ( 1 2 , 1)] chiral multiplet component in the nucleon, whose presence has been known for some time, but that had not been properly taken into account, yet. We have now extended these calculations of ΣπN to three light flavours, i.e., to SUL(3) × SUR(3) multiplet mixing. Phenomenology of chiral SUL(3) × SUR(3) multiplet mixing demands the presence of three chiral SUL(3) × SUR(3) multiplets, viz. [(6, 3) ⊕ (3, 6)], [(3, 3) ⊕ ( 3, 3)] and [( 3, 3) ⊕ (3, 3)], in order to successfully reproduce the baryons' flavor-octet and flavor-singlet axial currents, as well as the baryon anomalous magnetic moments. Here we use these previously obtained results, together with known constraints on the explicit chiral symmetry breaking in baryons to calculate the ΣπN term, but find little, or no change of ΣπN from the above successful two-flavor result. The physical significance of these results lies in the fact that they show no need for q 4 q components, and in particular, no need for an ss component in the nucleon, in order to explain the large "observed" ΣπN value. We also predict the experimentally unknown kaon-nucleon sigma term ΣKN .