The equilibration of Ising systems in random magnetic fields at low temperatures ($T$) following a quench from high $T$ is studied in the framework of a simple solid-on-solid model. The rate at which ordered domains in the model grow with time in any dimension ($d$) is estimated as a function of the random-field strength, the exchange strength, and $T$, on the basis of an approximate analogy to the problem of a one-dimensional random walk in a random medium. Domains are argued to grow logarithmically with time for all $d$. This result has a simple interpretation in terms of energy barriers which must be surmounted in the equilibration process.

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