In the context of a conjecture of Erdős and Gyárfás, we consider, for any $q ≥ 2$, the existence of q-power cycles (i.e. with length a power of q) in cubic graphs. We exhibit constructions showing that, for every $q ≥ 3$, there exist arbitrarily large cubic graphs with no q-power cycles. Concerning the remaining case $q = 2$ (which corresponds to the conjecture of Erdős and Gyárfás), we show that there exist arbitrarily large cubic graphs whose only 2-power cycles have length 4 only, or 8 only.