Following a problem posed by Lovasz in 1969, it is believed that every connected vertex-transitive graph has a Hamilton path. This is shown here to be true for cubic Cayley graphs arising from groups having a $(2,s,3)$-presentation, that is, for groups $G=la a,b| a^2=1, b^s=1, (ab)^3=1, etc. ra$ generated by an involution $a$ and an element $b$ of order $sgeq3$ such that their product $ab$ has order 3. More precisely, it is shown that the Cayley graph $X=Cay(G,{a,b,b^{-1}})$ has a Hamilton cycle when $|G|$ (and thus $s$) is congruent to 2 modulo 4, and has a long cycle missing only two vertices (and thus necessarily a Hamilton path) when $|G|$ is congruent to 0 modulo 4.
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