It is a challenging task to design a random access protocol that achieves the optimal throughput in multi-cell random access with decentralized transmission due to the difficulty of coordination. In this paper, we present a decentralized interference-aware opportunistic random access (IA-ORA) protocol that enables us to obtain the optimal throughput scaling in an ultra-dense multi-cell random access network with one access point (AP) and a number of users. In sharp contrast to opportunistic scheduling for cellular multiple access where users are selected by base stations, under the IA-ORA protocol, each user opportunistically transmits with a predefined physical layer (PHY) data rate in a decentralized manner if not only the desired signal power to the serving AP is sufficiently large but also 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 shown that the optimal aggregate 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 exceeds some level. Additionally, it is numerically demonstrated via computer simulations that under practical settings, the proposed IA-ORA protocol outperforms conventional opportunistic random access protocols in terms of aggregate throughput.