  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 $\Pi _{C}\mathcal{M}%
\vDash \urcorner \Phi .$
