We consider asymmetric cyclic polling systems with an arbitrary number of queues, general service-time distributions, zero switch-over times, gated service at each queue, and with general renewal arrival processes at each of the queues. For this classical model, we propose a new method to derive closed-form expressions for the expected delay at each of the queues when the load tends to 1, under proper heavy-traffic (HT) scalings. In the literature on polling models, rigorous proofs of HT limits have only been obtained for polling models with Poisson-type arrival processes, whereas for renewal arrivals HT limits are based on conjectures [E.G. Coffman, A.A. Puhalskii, M.I. Reiman, Polling systems with zero switch-over times: A heavy-traffic principle, Ann. Appl. Probab. 5 (1995) 681–719; E.G. Coffman, A.A. Puhalskii, M.I. Reiman, Polling systems in heavy-traffic: A Bessel process limit, Math. Oper. Res. 23 (1998) 257–304; T.L. Olsen, R.D. van der Mei, Periodic polling systems in heavy-traffic: Renewal arrivals, OR Lett. 33 (2005) 17–25]. Therefore, the main contribution of this paper lies in the fact that we propose a new method to rigorously prove HT limits for a class of non-Poisson-type arrivals. The results are remarkably simple and provide new fundamental insight and reveal explicitly how the expected delay at each of the queues depends on the system parameters, and in particular on the interarrival-time distributions at each of the queues. The results also suggest simple approximations for the expected delay in stable polling systems. Numerical results show that the approximations are highly accurate when the system load is roughly 90% or more.