Numerical simulations of the early transition process in periodic grooved‐channel flow are presented. For Reynolds numbers,R<Rc,1=O(100), the two‐dimensional steady flow is stable to all disturbances; atR=Rc,1the flow undergoes a supercritical Hopf bifurcation to a nonlinear two‐dimensional steady‐periodic state; forR>Rc,2>Rc,1the wavy two‐dimensional flow is unstable to a classical linear three‐dimensional secondary instability; and for some range of Reynolds number aboveRc,2the secondary instability saturates in a steady‐periodic, three‐dimensional, low‐order equilibrium. The three‐dimensional equilibria owe their existence and stability to thenarrowbandnature of grooved‐channel‐flow secondary instability, which in turn reflects the low‐Reynolds‐numbersupercriticalform of the grooved‐channel‐flow primary bifurcation. The contrast between the low‐order, weak transition in ‘‘inflectional’’ complex‐geometry channels and the abrupt, snap‐through transition in (subcritical‐primary broadband‐secondary) planar channels illustrates the important role of primary criticality in the early transition process.