STT-MRAM is regarded as an extremely promising NVM technology for replacing SRAM-based on-chip memories. While STT-MRAM memories benefit from ultra-low leakage power and high density, they suffer from some reliability challenges, namely, read disturbance, write failure, and retention failure. The write failure; storing a wrong value in an STT-MRAM cell during a write operation, is the most crucial reliability challenge. In this article, we propose ROCKY; a robust architecture equipped with efficient replacement policies for STT-MRAM-based cache memory hierarchy to improve the robustness of STT-MRAM part against the write failures. ROCKY reduces susceptible transitions in STT-MRAM cache memories leading to more reliable STT-MRAM write operations. The simulation results through comparison with traditional cache memory hierarchy demonstrate ROCKY decreases the WER of STT-MRAM cache memories by up to 35.4 percent while imposing less than 1 percent performance overhead to the system.