As I just remembered, the answer is yes if $X$ is quasiseparated, noetherian and normal: see Laumon and Moret-Bailly, Champs algébriques, (16.6.2). The quesiseparated assumption is needed (see Scott Carnahan's answer). Without the normality condition, I don't have a counterexample but my guess is that there is one.
Assume $X$ noetherian, integral, normal and, to simplify, separated (I am not sure how much this helps). Cover $X$ by étale maps $X_i\to X$ with each $X_i$ integral and affine. There is a dense open subspace $U$ of $X$ which is a scheme and such that each induced map $U_i:=X_i\times_X U\to U$ is finite. Let $V\to U$ be an étale Galois cover, with Galois group $G$, dominating all the $U_i$'s. Now let $\overline{V}\to X$ (resp. $\overline{X_i}\to X$) be the normalization of $X$ in $V$ (resp. in $X_i$); we have dense open immersions $V\subset \overline{V}$ and $X_i\subset \overline{X_i}$. By functoriality, $G$ acts on $\overline{V}$, with quotient $X$.
I claim that $\overline{V}$ is a scheme. Indeed, for each $i$, $\overline{X_i}$ is also the normalization of $X$ in $U_i$. In particular there is an $X$-morphism $f_i:\overline{V}\to\overline{X_i}$ (deduced from $V\to U_i$) which must be finite surjective (everyone is integral, finite and surjective over $X$). Put $V_i:=f_i^{-1}(X_i)$: this is an open subspace of $\overline{V}$ which is finite over $X_i$, hence an affine scheme. So, the union $W$ of the $V_i$'s is an open subspace of $\overline{V}$ which is a scheme and maps surjectively to $X$ (since $V_i\to X_i$ is surjective), hence $X$ \overline{V}$is covered by $\{gW\}_{g\in G}$. 2 added 1518 characters in body; added 33 characters in body EDIT after Chris' and Jason's comments: in fact the proof in the case of normal algebraic spaces can be made substantially simpler than in the book (which proves a more general result about noetherian Deligne-Mumford stacks). It goes like this: Assume$X$noetherian, integral, normal and, to simplify, separated (I am not sure how much this helps). Cover$X$by étale maps$X_i\to X$with each$X_i$integral and affine. There is a dense open subspace$U$of$X$which is a scheme and such that each induced map$U_i:=X_i\times_X U\to U$is finite. Let$V\to U$be an étale Galois cover, with Galois group$G$, dominating all the $U_i$'s. Now let$\overline{V}\to X$(resp.$\overline{X_i}\to X$) be the normalization of$X$in$V$(resp. in$X_i$); we have dense open immersions$V\subset \overline{V}$and$X_i\subset \overline{X_i}$. By functoriality,$G$acts on$\overline{V}$, with quotient$X$. I claim that$\overline{V}$is a scheme. Indeed, for each$i$,$\overline{X_i}$is also the normalization of$X$in$U_i$. In particular there is an$X$-morphism$f_i:\overline{V}\to\overline{X_i}$(deduced from$V\to U_i$) which must be finite surjective (everyone is integral, finite and surjective over$X$). Put$V_i:=f_i^{-1}(X_i)$: this is an open subspace of$\overline{V}$which is finite over$X_i$, hence an affine scheme. So, the union$W$of the$V_i$'s is an open subspace of$\overline{V}$which is a scheme and maps surjectively to$X$(since$V_i\to X_i$is surjective), hence$X$is covered by $\{gW\}_{g\in G}$. 1 As I just remembered, the answer is yes if$X\$ is quasiseparated, noetherian and normal: see Laumon and Moret-Bailly, Champs algébriques, (16.6.2). The quesiseparated assumption is needed (see Scott Carnahan's answer). Without the normality condition, I don't have a counterexample but my guess is that there is one.