CHICAGO, July 21 (Reuters) - A U.S. appeals court on Tuesday vacated five out of 18 criminal convictions against former Illinois Governor Rod Blagojevich, who is serving a 14-year sentence for attempted extortion from campaign contributors, wire fraud and other crimes.

The judges vacated his sentence and ordered a retrial on the five counts, but said Blagojevich, 58, was not entitled to release from prison pending retrial because most convictions were upheld and the advisory sentencing range is above 14 years.

An attorney for Blagojevich said he believed his client should pursue further appeals and Blagojevich's wife, Patti Blagojevich, told a news conference her husband is an innocent man who never intended to break the law.

"He's optimistic that justice will prevail, eventually," Patti Blagojevich said, adding that he was disappointed by the decision and hopeful of a reduction in sentence.

Blagojevich, a Democrat, was arrested in December 2008 when he was still governor. He was impeached by the state's General Assembly in early 2009, becoming the first Illinois governor to be removed from office.

The 18 convictions came in two jury trials. Blagojevich began serving his federal prison sentence in March 2012.

At the heart of the convictions were Blagojevich's attempts in 2008 to make money out of his power to appoint a replacement for Barack Obama , who was leaving his seat in the U.S. Senate representing Illinois after winning the presidential election.

In the appeal to the 7th Circuit U.S. Court of Appeals, Blagojevich's lawyers argued the evidence was insufficient to convict him on any count. Judge Frank Easterbrook called that argument "frivolous."

"The evidence, much of it from Blagojevich's own mouth, is overwhelming," Easterbrook wrote in a 23-page opinion.

However, the judges found a problem in instructions to the jury on counts that concerned Blagojevich's proposal to appoint senior Obama adviser Valerie Jarrett to the Senate seat in exchange for a cabinet appointment for himself, requiring those convictions to be vacated.

Leonard Goodman, an attorney for Blagojevich, said the appeals court either did not address key points or erred.

"He never put a penny in his pocket," Goodman said. "He bought his own Cubs tickets. He bought his own clothes. He never took personal favors from campaign contributors." (Additional reporting by Emily Stephenson; Editing by Sandra Maler and Mohammad Zargham)