Final results gave the former Massachusetts governor 39 percent of the vote, compared with 32 percent for the former senator from Pennsylvania. U.S. Rep. Ron Paul of Texas garnered 21 percent in the straw poll and former House Speaker Newt Gingrich got 8 percent, the party said.
The precinct party caucuses make up the first event in Wyoming's lengthy political system, Wyoming Public Radio reported.
"The caucus is great," says Khale Lenhart, vice chairman of the Laramie County Republicans, "because it allows people a chance to come out and debate on the local level among their neighbors, and gives people a chance to participate and actually be involved in a way that allows them to advocate and express their support."
The precinct caucuses also elect delegates to county Republican conventions, which are scattered through early March, said Tammy Hooper, head of the state GOP.
She said 12 delegates to the national convention and 12 alternates are to be picked from Tuesday to Saturday. In mid-April, 14 at-large presidential delegates will be selected during the statewide Republican convention.
The state has 29 non-binding delegates.
Wyoming's process takes more than two months, but caucus-goers like Kim Deti tell Wyoming Public Radio they like the pace.
"It's the building block of the whole process -- you have to have a foundation for a system that's going to work, and this really is a purely democratic foundation," Deti said, adding that anyone can show up and end up in Tampa.
Russ Donley of Casper says he's been involved in the GOP since college and now is a precinct committeeman. He told the Casper Star Tribune he'd be happy with any of the presidential candidates but thinks Mitt Romney would have the resources to compete with President Obama.
"Our job is to help get Republicans elected," he said.