jats:pThe stability of steady convective exchange flow with a rectangular planform in an unbounded three-dimensional porous medium is explored. The base flow comprises a balance between vertical advection with amplitude jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112017005614_inline1" />jats:tex-math$A$</jats:tex-math></jats:alternatives></jats:inline-formula> in interleaving rectangular columns with aspect ratio jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112017005614_inline2" />jats:tex-math$𝜉⩽1$</jats:tex-math></jats:alternatives></jats:inline-formula> and horizontal diffusion between the columns. Columnar flow with a square planform (jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112017005614_inline3" />jats:tex-math$𝜉=1$</jats:tex-math></jats:alternatives></jats:inline-formula>) is found to be weakly unstable to a large-scale perturbation of the background temperature gradient, irrespective of jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112017005614_inline4" />jats:tex-math$A$</jats:tex-math></jats:alternatives></jats:inline-formula>, but to have no stronger instability on the scale of the columns. This result provides a stark contrast to two-dimensional columnar flow (Hewitt jats:italicet al.</jats:italic>, jats:italicJ. Fluid Mech.</jats:italic>, vol. 737, 2013, pp. 205–231), which, as jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112017005614_inline5" />jats:tex-math$A$</jats:tex-math></jats:alternatives></jats:inline-formula> is increased, is increasingly unstable to a perturbation on the scale of the columnar wavelength. For rectangular planforms with jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112017005614_inline6" />jats:tex-math$\backslash unicode[STIX]\{x1D709\}<1$</jats:tex-math></jats:alternatives></jats:inline-formula>, a critical aspect ratio is identified, below which a perturbation on the scale of the columns is the fastest growing mode, as in two dimensions. Scalings for the growth rate and the structure of this mode are identified, and are explained by means of an asymptotic expansion in the limit jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112017005614_inline7" />jats:tex-math$𝜉\to 0$</jats:tex-math></jats:alternatives></jats:inline-formula>. The difference between the stabilities of two-dimensional and three-dimensional exchange flow provides a potential explanation for the apparent difference in dominant horizontal scale observed in direct numerical simulations of two-dimensional and three-dimensional statistically steady ‘Rayleigh–Darcy’ convection at high Rayleigh numbers.</jats:p>