Odierno's comments came as a controversy raged in Iraq over a government spokesman's suggestion that US forces might not fully withdraw by the end of 2011.
Ali al-Dabbagh said on a visit to Washington this week that Iraqi security forces might need 10 years to get ready to take over from US troops.
The office of Nuri al-Maliki, the Iraqi prime minister, dismissed his remarks, saying: "What Dr Ali al-Dabbagh announced about Iraqi forces needing 10 years to be ready was a personal opinion and does not represent the Iraqi government opinion."
Nazzal Makki, an Iraqi journalist based in Amman, told Al Jazeera the US was going through a partial withdrawal from Iraq.
"We expect that the final step will involve the keeping of some forces in huge bases in Iraq because that's what they were built and prepared for," he said.
"Anyone who thought that the upcoming months might present opportunities to test the new administration would be sorely mistaken"
US defence secretary
"However, we hope Obama will have the wisdom to withdraw completely from Iraq - that's our wish.
"Over the short term, Americans aren't doing much in maintaining security, so the withdrawal of a few thousand soldiers won't bring much change."
Makki added: "Let's hope the US army has the wisdom to prepare Iraq for ruling itself, otherwise we see no sign of them doing that."
Under the terms of a US-Iraq pact that comes into effect in January, US forces must leave Iraq by the end of 2011. They must leave urban areas by June 2009.
Iraq has seen a decline in the overall level of violence that came in the wake of the 2003 US-led invasion.
But officials say that attacks could increase in the run-up to provincial and general elections next year.
There are also concerns that opposition fighters could launch attacks as US forces begin to leave the country's town and cities.
Gates, who will retain his position in president-elect Barack Obama's administration, said that the United States' enemies should not try to take advantage of the early stages of the incoming government.
Before travelling to Iraq, he told a security conference in Bahrain that the transition from George Bush, the outgoing president, to Obama had been well planned.
"Anyone who thought that the upcoming months might present opportunities to test the new administration would be sorely mistaken," he told the Manama Dialogue conference.
Gates also assured delegates that the US would continue to be involved in the Middle East after Obama becomes president in January.
"I can assure you that a change in administration does not alter our fundamental interests, especially in the Middle East," he said.
Joe Biden, the US vice-president elect, has already said that countries opposed to the US could try to spark a crisis early in Obama's administration.
But Gates told the conference: "President Obama and his national security team, myself included, will be ready to defend the interests of the United States and our friends and allies from the moment he takes office on January 20."
No 'regime change'
Gates also called on Sunni Arab states to support Iraq's US-backed government by committing to full diplomatic relations with Baghdad and by writing off Iraqi debts run up during the rule of Saddam Hussein, the late president.
Most Sunni Arab countries, including Saudi Arabia, have in the past said that the Iraqi government is too close politically to Iran, whose government is Shia Muslim.
"There is no doubt that Iran has been heavily engaged in trying to influence the development and direction of the Iraqi government - and has not been a good neighbour," Gates said.
"Iraq wants to be your partner ... And, given the challenges in the Gulf, and the reality of Iran, you should wish to be theirs."
Gates told the conference that Washington was not looking to remove Iran's leaders but it wants Tehran to make changes.
"Nobody is after a regime change in Iran. What we're after is a change in policies and a change in behaviour.
"The president-elect and his team are under no illusions about Iran's behaviour and what Iran has been doing in the region and is doing in terms of its own weapons programmes."
An Iranian delegation had been due to attend the meeting but did not show up for Gates's address.