Cohesive powers of computable structures can be viewed as effective ultraproducts over effectively indecomposable sets called cohesive sets. We investigate the isomorphism types of cohesive powers $Pi _{C}% mathcal{L}$ for familiar computable linear orders $mathcal{L}$. If $% mathcal{L}$ is isomorphic to the ordered set of natural numbers $mathbb{N}$ and has a computable successor function, then $Pi _{C}mathcal{L}$ is isomorphic to $mathbb{N}+mathbb{Q}times mathbb{Z}.$ Here, $+$ stands for the sum and $times $ for the lexicographical product of two orders. We construct computable linear orders $mathcal{L}_{1}$ and $mathcal{L}_{2}$ isomorphic to $mathbb{N},$ both with noncomputable successor functions, such that $Pi _{C}mathcal{L}_{1}mathbb{ }$is isomorphic to $mathbb{N}+% mathbb{Q}times mathbb{Z}$, while $Pi _{C}mathcal{L}_{2}$ is not$.$ While cohesive powers preserve all $Pi _{2}^{0}$ and $Sigma _{2}^{0}$ sentences, we provide new examples of $Pi _{3}^{0}$ sentences $Phi $ and computable structures $% mathcal{M}$ such that $mathcal{M}vDash Phi $ while $Pi _{C}mathcal{M}% vDash urcorner Phi .$
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