Such simple groups are a subset of the finite simple "thin" groups, and the latter have been classified (by Michael Aschbacher in 1976 and 1978). A $2$-local subgroup of a finite group is the normalizer of some non-trivial $2$-subgroup. A finite simple group $G$ is said to be "thin" if all its $2$-local subgroups have cyclic Sylow subgroups for all odd primes.

We may note that Feit and Thompson proved early in their odd order paper (using transfer theorems) that if $G$ is a finite group of odd order which has no elementary Abelian subgroup $p$-subgroup of rank $3$ or more for any prime $p$, then $G$ has a normal Sylow $q$-subgroup where $q$ is the largest prime divisor of $|G|$. Hence (without using the full force of the odd order theorem), it is indeed the case that if a finite non-Abelian simple group $G$ has cyclic Sylow subgroups for all odd primes $p$, then $G$ has even order, and is, in particular, a thin group. However, some thin groups do not satisfy the condition of the question.

According to Aschbacher's classification, the list of thin finite simple groups is ${\rm SL}(2,2^{n})$ and ${\rm PSL}(2,q)$ for $q$ an odd prime power (but we can remove the cases when the odd $q$ is not prime in answer to the present question since the Sylow subgroup of order $q$ is not cyclic), ${\rm PSL}(3,4)$ (but we can remove this in answer to the present question since it has a non-cyclic subgroup of order $9$), ${\rm PSL}(3,p)$ for $p$ of the form $1+2^{a}3^{b}$ (which can be removed in answer to the present question when $p$ is odd, since there are non-cyclic $p$-subgroups in that case), ${\rm PSU}(3,p)$ ($p = 2^{a}3^{b}-1$, $b \in \{0,1\}$ (but these can be omitted in answer to the present question as there are non-cyclic $p$-subgroups), ${\rm PSU}(3,2^{n})$ (but these can be omitted in answer to the present question, since ${\rm PSU}(3,4)$ has an elementary Abelian subgroup of order $25$), ${\rm Sz}(2^{n})$ , the Tits group $^{2}F_{4}(2)^{\prime}$ (which can be omitted in answer to the present question, as it contains an elementary Abelian subgroup of order $9$), the Steinberg triality group
$^{3}D_{4}(2)$ (which can be omitted in answer to the present question since it has elementary Abelian subgroups of order $9$), the Mathieu group $M_{11}$ (which can be omitted in answer to the present question as it has elementary Abelian subgroups of order $9$), and the first Janko group $J_{1}.$

Later edit: Notice that Aschbacher's classification of the thin finite simple groups predates the full classification of the finite simple groups. As a matter of historical interest, the classification of finite simple groups in which all $2$-local subgroups have low $p$-rank for all odd primes $p$ is a difficult part of the current proofs (and of ongoing revisions) of the classification of finite simple groups.