For cyclic groups $C_p$ of prime order $p$ the irreducible integral representations are known (I don't know if there are results for cyclic groups of composite order but it's likely since the result for prime orders is very old):

Let $\zeta \in \mathbb{C}$ be a $p$-th root of unity and let $h_p$ be the class number of $\mathbb{Q}(\zeta)$. Up to isomorphism there are exactly $1+2h_p$ irreducible $\mathbb{Z}C_p$-modules.

If $B_1,...,B_h \subseteq \mathbb{Q}(\zeta)$ are fractional ideals that represent the elements of the class group of $\mathbb{Q}(\zeta)$ und $b_i \in B_i \setminus (1-\zeta)B_i$, then the irreducible $\mathbb{Z}C_p$-modules are:
$$\mathbb{Z},\,\,B_i,\,\,B_i\oplus \mathbb{Z}\qquad(i=1,...,h)$$
If $\sigma$ is a generator of $C_p$ then the action is given by $\sigma \cdot b = \zeta b, \, \sigma\cdot (b,1) = (\zeta b+b_i,1)$.

*Reference:* Curtis, Reiner: Representation Theory of Finite Groups and Associative Algebras, Theorem (74.3) and Exercise 4 in § 74.9.

*Note:* These irreducible representations depend on the class number of cyclotomic fields. Since this class number is not known in general there is no explicit list in general.