Washington (CNN) -- President Barack Obama on Monday nominated Gen. Martin Dempsey, the relatively new Army chief of staff, to be the next chairman of the Joint Chiefs of Staff.

The term of the current chairman, Adm. Michael Mullen, ends in September.

Obama also nominated Adm. James Winnefeld, the current head of the U.S. Northern Command, to serve as vice chairman. He tapped Gen. Ray Odierno to replace Dempsey as the Army's chief of staff.

The president, speaking at the White House, urged the Senate to confirm the three men "as swiftly as possible" in order to help ensure a "seamless" Pentagon transition.

Who is Gen. Martin Dempsey?

Obama praised Dempsey as one of the country's most "respected and combat-tested generals."

"I expect him to push all our forces to keep adapting and innovating," Obama said. "We will provide what it takes to achieve our objectives in the current fight."

Outgoing Defense Secretary Robert Gates released a statement praising Dempsey, Winnefeld, and Odierno for having "excelled in key command and staff roles" and possessing "the right mix of intellectual heft, moral courage, and strategic vision."

If confirmed by the Senate, Dempsey would be in the key position of being Obama's senior military adviser overseeing the drawdown of troops in Iraq and Afghanistan. He would also work with incoming Defense Secretary Leon Panetta, who still needs to be confirmed by the Senate, in establishing priorities for cutting defense spending.

Dempsey has extensive combat experience. He served as the commander of the 1st Armored Division in Baghdad in 2003, and then helped train Iraqi security forces in a subsequent tour of duty in Iraq. He also has served as acting commander of the U.S. Central Command overseeing U.S. military operations in the Middle East, Persian Gulf and Central Asia.

The selection of Dempsey is unusual partly because he took over as head of the Army only on April 11. In that brief period of time, Dempsey has earned strong praise for what is considered to be his outgoing personality. Among other things, the general receives a list of soldiers both in the United States and overseas every week and calls them to get their direct opinions on life in the Army.

He spent last Friday in Washington meeting with children who have lost parents and other relatives in war-time military service.

He has also gained attention for his long-standing habit of beginning some of his public speaking events by singing songs such as "New York, New York." On his Facebook page he recently asked troops to offer their suggestions for music he could play at events where he is speaking.

Dempsey's sense of humor has opened him up for some ribbing as well, like at the ceremony when he took command of the Army.

"His rise to this great height is yet another one of those classic American immigrant success stories," said Army Secretary John McHugh during official remarks. "One can only imagine how different his life might have been had his family not decided to leave New Jersey and move across the river to New York state."

Gen. James Cartwright, a four-star Marine who currently is vice chairman of the Joint Chiefs, had been the perceived front runner to replace Mullen. But in a 30-minute meeting with Obama before the president left for his recent trip to Europe, Cartwright was told he would not get the job, according to a Defense Department official with direct knowledge of the meeting.

The official declined to be identified because of the sensitivity of discussing White House personnel meetings involving the president.

Cartwright is now expected to submit his retirement papers and conclude his term as vice chairman as scheduled in early August.

Cartwright had long been considered a White House favorite, but had run afoul of other senior officers inside the Pentagon who felt he more than once went behind their backs in offering the president military recommendations and advice on a range of issues, including lower troop levels in Afghanistan.

The Defense Department official who is familiar with Cartwright's thinking says the general was aware of these criticisms, and only offered recommendations when asked by the president.

"There was an impression created he wasn't a team player," said the official, who was offering a defense of Cartwright's actions

"Gen. Cartwright never believed he was a deputy. He believed he had a legal and moral obligation to give the president his best military advice," the source said. "He is not obligated to provide the same advice as everybody else."

The official acknowledged the widespread view many had, however, that Cartwright plays his cards close to the vest inside the Pentagon and often shared little with colleagues about his real thinking.

The official also noted that neither Gates nor Mullen had recommended Cartwright for the job as chairman, which meant his nomination by the White House would have been difficult to justify to Congress.

Spokesmen for both declined to publicly comment on any aspect of the chairmanship issue.

Cartwright had also been the subject of an investigation by the department's inspector general over whistleblower allegations of improper conduct with a junior female officer who fell asleep intoxicated in his hotel room during a business trip. He was cleared of wrongdoing by Navy Secretary Ray Mabus, but the official said Cartwright felt he was the still the subject of a "whispering campaign" after that.

CNN's Alan Silverleib and Josh Levs contributed to this report.