Speed, who had a wife and two children, was found hanged at his home on Sunday.
An inquest into the death, which is not being treated as suspicious, is due to begin on Tuesday.
"We have been overwhelmed by the support and it really has helped," his agent, Hayden Evans, said outside the family's home in Cheshire.
"Gary's family would sincerely like to thank all the people that have sent messages of condolence and tributes in what is a very difficult time.
"We would ask that the family are now given the respect of some privacy to just grieve on their own."
Speed enjoyed a long and successful playing career that included spells at Leeds United, Everton, Newcastle United, Bolton Wanderers and Sheffield United.
He made 85 appearances for Wales between 1990 and 2004 and was named his country's manager last December.
Although they failed to qualify for the Euro 2012 finals, Wales had shown signs of improvement under Speed's guidance and had won their last three matches.
FIFA President Sepp Blatter was among those to pay tribute to Speed on Monday.
"Gary Speed was a hugely talented player and a great servant for both club and country who will be greatly missed," Blatter said in a letter to Football Association of Wales president Phil Pritchard.
"I would like to offer my deepest condolences on behalf of FIFA and the worldwide family of football to you, the Football Association of Wales, his wife Louise and their two sons, as well as Gary Speed's friends and family.
"I hope the knowledge that we are all thinking of them can provide some solace in this time of deep sadness."
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