In this article, a novel random pre-coding (RPC) based random access (RA) scheme is proposed to improve the system performance in the Internet of Things-oriented satellite networks. In this scheme, each device randomly selects a non-orthogonal multi-level complex sequence from a pre-defined pre-coding sequence set to pre-code the transmitted packets in a sub-frame, by which multiple superimposed packets in the sub-frame can be successfully decoded by using the minimum mean squared error (MMSE) based successive interference cancellation (SIC) detection algorithm. Besides, by combining the proposed RPC with the existing contention resolution diversity slotted ALOHA (CRDSA) scheme, which the RPC-CRDSA scheme is generated. The SIC processing should be carried out across sub-frames to remove the received signals that are already recovered, which can further significantly improve the system throughput. The performance of the RPC-RA scheme is evaluated via both theoretical analysis and computer simulation. The collision probabilities of both preamble and pre-coding sequences and an upper bound of throughput are derived, respectively. Simulation results show that the peak normalized throughput of RPC-CRDSA with the pre-coding sequence length $L = 8$ can reach 1.87 bits/symbol, which largely outperforms traditional schemes.