Due to the difficulty of coordination in multi-cell random access, it is a practical challenge how to achieve the optimal throughput with decentralized transmission. In this paper, we propose a decentralized interference-aware opportunistic random access (IA-ORA) protocol that achieves the optimal throughput scaling in an ultra-dense multi-cell random access network with one access point (AP) and a number of users in each cell. Unlike opportunistic scheduling for cellular multiple access where users are selected by base stations, under our IA-ORA protocol, each user opportunistically transmits with a predefined physical layer data rate in a decentralized manner if the desired signal power to the serving AP is sufficiently large and the generating interference leakage power to the other APs is sufficiently small (i.e., two threshold conditions are fulfilled). As a main result, it is proved that the optimal aggregate physical layer throughput scaling (i.e., the MAC throughput of $\frac{1}{e}$ in a cell and the power gain) is achieved in a high signal-to-noise ratio regime if the number of per-cell users is higher than a certain level. In addition, numerical evaluation via intensive computer simulations confirms that under practical settings, the proposed IA-ORA protocol outperforms conventional opportunistic random access protocols in terms of aggregate throughput.