Translocation through nanopores has emerged as a new experimental technique to probe the physical properties of biomolecules. The question of how the typical translocation time for a single unstructured polymer depends on its length has already triggered many theoretical and computational studies. Here, we address the same question, but for structured RNA molecules where the breaking of base-pairing patterns is the main barrier for translocation. Within a simple model, we calculate the typical time for single-stranded RNAs with random sequences to translocate through an idealized nanopore. It is believed that with respect to secondary structure formation, such a random RNA is typically either frozen into a single dominant structure (glassy phase) or many different structures with similar total energies coexist (molten phase), depending on the temperature and the base-pairing energetics. We find that these two phases can be clearly distinguished by their translocation behavior in the absence of an external voltage bias. In both cases, the typical translocation time depends on the sequence length as a power law with an exponent that exceeds the naively expected exponent of two for purely diffusive translocation. However, whereas in the molten phase the exponent is constant, at a value of 5/2, the exponent increases rapidly with decreasing temperature in the glassy phase. We explain the behavior in the molten phase and the qualitative trend in the glassy phase theoretically.