This paper discusses a tri-layer non-cooperative energy trading approach among multiple grid-tied multi-energy microgrids (MEMGs) in the restructured integrated energy market. The heterogeneous uncertainties from renewable energy, market prices, and electric energy loads are also considered via the risk-averse stochastic programming (SP) approach. First, comprehensive operation models of individual MEMGs are presented with the consideration of practical electric energy and thermal network flows as well as battery degradation. Second, to guarantee fair multi-energy trading among MEMGs and deal with adverse effects from all uncertainty sources, a tri-layer Cournot Nash game-based energy bidding method is developed and solved by the SP approach. In the first layer, i.e., day-ahead multi-energy market, optimal energy bids, dispatches of energy storage assets, and thermal flows against uncertainty scenarios are acquired in a risk-averse manner; In the second layer, i.e., intra-day multi-energy market, optimal intra-day energy bids and dispatches of all resources against uncertainty realizations are sequentially calculated; In the third layer, i.e., the real-time multi-energy market, transactions between each MEMG and the wholesale multi-energy market are finalized. Third, for protecting the privacy of individual MEMGs and alleviating the computation burdens, the distributed alternating search procedure is employed to compute the Nash equilibriums in the day-ahead and intra-day markets. In the end, numerical case studies are conducted to verify the effectiveness of our method. From the simulation results, it can be inferred that compared with the traditional cooperative, deterministic and risk-natural methods in the literature, our proposed method is more practical and economical for real-world applications since it comprehensively considers the market competition, uncertainty handling, and energy trading risk.