CAIRO, June 20 (Reuters) - The U.S. ambassador to Egypt is under fire from opposition groups who were angered by her criticism this week of planned mass rallies against the ruling Muslim Brotherhood.

Anne Patterson responded to widespread talk among Egyptian liberals that Washington had thrown its weight behind Islamist President Mohamed Mursi by saying in a speech on Tuesday that the United States was working closely with the elected government and also listening to allEgypt's political groups.

But by Thursday, after extensive local media coverage of her remarks and condemnation by opposition leaders of "interference" in Egypt's internal affairs, social media was dominated by angry and hostile comments directed toward Patterson and her embassy.

Among the more polite, tycoon Naguib Sawiris tweeted: "Madam Ambassador, ... Please bless us with your silence."

Groups involved in promoting demonstrations for June 30 aimed at pushing Mursi to step down a year after he took office, rejected her suggestion they risked further violence and would be better engaged in improving their electoral organisations.

"This is the government that you and your fellow citizens elected," she said. "Even if you voted for others, I don't think the elected nature of this government is seriously in doubt.

"The United States took the position that we would work with whoever won elections that met international standards."

Of the planned protests, she said: "Egypt needs stability to get its economic house in order, and more violence on the streets will do little more than add new names to the lists of martyrs. Instead, I recommend Egyptians get organised.

"Join or start a political party that reflects your values and aspirations. Egyptians need to know a better path forward. This will take time. You will have to roll up your sleeves and work hard."

The United States forged an alliance with Egypt's military rulers during the Cold War, helping persuade them to sign a peace treaty with Israel. It provided substantial aid throughout the 30 years that Hosni Mubarak ran the country, despite qualms about a lack of democracy. It continues to provide aid.

Patterson also spoke of what she called "conspiracy theories" that Washington connived to topple Mubarak in 2011 and replace him with the Brotherhood, with which U.S. diplomats had long maintained contact while it was banned and in opposition.

"Such speculation is groundless," she said. (Reporting by Alastair Macdonald; Editing by Michael Roddy)