(CNN) Donald Trump won last month's Republican primary in Georgia but Ted Cruz made inroads in the state over the weekend.

Each of Georgia's 14 congressional districts held conventions on Saturday to choose delegates to send to the Republican National Convention. After they met, it became clear that many delegates would vote for Cruz if the convention in Cleveland goes to multiple ballots.

Delegates

Trump won almost 40% of the vote and came in first in all but four counties in Georgia's primary. By state rules, 42 delegates are therefore "bound" to support him on the first round of voting at the national convention. However, if no nominee is chosen at the first ballot, these delegates will no longer be bound to vote for Trump and instead may support another candidate.

It was clear after Saturday's delegate selection that many of these delegates, once "unbound," would vote for Cruz.

The Texas senator came in third by popular vote in Georgia, and won no counties in the state, but he has earned 18 pledged Georgia delegates, who will be "bound" for him on the first ballot. After organizing at the county level in March and again at the congressional districts on Saturday, Cruz's supporters make up a large portion of Georgia's convention delegation so far.

The delegate fight in Georgia is not yet over, with 31 more Republican delegates -- the majority of whom will be bound to vote for Trump on the convention's first ballot -- to be selected at a state convention in early June.

Controversial process

Many Trump supporters cried foul at the process, agreeing with their preferred candidate that the system was "rigged."

Georgia's third district selected a slate of national delegates openly in opposition to the billionaire's candidacy after a somewhat contentious show of support for Trump.

Pam Ausman, a Trump supporter from the district, said, "Why are we voting if they don't listen to us?"

Brant Frost, a prominent Republican and Cruz supporter from the district, vehemently denied such charges.

"These are the moms and dads of heartland America, playing by the rules," she said.

Katie Frost, the elder Frost's daughter and a prominent Republican in her own right, was among those selected in the third district. She came to the convention with copies of a letter from Cruz supporting her bid for a delegate spot along with a picture of the two skeet shooting.

She also disputed Trump's characterization of the delegate process as "rigged."

"(Trump) doesn't understand the process," she said.

Several Trump supporters, one carrying an American flag, left the convention in Georgia's seventh district after Trump supporter and conservative activist Debbie Dooley was not chosen as a delegate.

Elsewhere, anger at the process was somewhat muted.

Brandon Phillips, Trump's Georgia campaign director, said he was content with the results of his home district's convention, which chose him as one of its three delegates.

Despite the overall success of Cruz, the Trump campaign pointed to a few victories and cast Saturday's contests as beating expectations going in.

"A majority of Mr. Trump's delegates selected today support Mr. Trump," Trump national delegate director Brian Jack told CNN on Saturday.

Ohio Gov. John Kasich won zero delegates in the Georgia primary election, but his campaign claimed four delegates in the state's district conventions Saturday, including State Sen. Bill Cowsert.