This paper studies the determination of third‐ and fourth‐order bounds on the effective conductivity σe of a composite material composed of aligned, infinitely long, identical, partially penetrable, circular cylinders of conductivity σ2 randomly distributed throughout a matrix of conductivity σ1. Both bounds involve the microstructural parameter ζ2 which is a multifold integral that depends upon S3, the three‐point probability function of the composite. This key integral ζ2 is computed (for the possible range of cylinder volume fraction ϕ2) using a Monte Carlo simulation technique for the penetrable‐concentric‐shell model in which cylinders are distributed with an arbitrary degree of impenetrability λ, 0≤λ≤1. Results for the limiting cases λ=0 (‘‘fully penetrable’’ or randomly centered cylinders) and λ=1 (‘‘totally impenetrable’’ cylinders) compare very favorably with theoretical predictions made by Torquato and Beasley [Int. J. Eng. Sci. 24, 415 (1986)] and by Torquato and Lado [Proc. R. Soc. London Ser. A 417, 59 (1988)], respectively. Results are also reported for intermediate values of λ: cases which heretofore have not been examined. For a wide range of α=σ2/σ1 (conductivity ratio) and ϕ2, the third‐order bounds on σe significantly improve upon second‐order bounds which just depend upon ϕ2. The fourth‐order bounds are, in turn, narrower than the third‐order bounds. Moreover, when the cylinders are highly conducting (α≫1), the fourth‐order lower bound provides an excellent estimate of the effective conductivity for a wide range of volume fracti- ons.