Keane pays tribute to Trapattoni
Keane, 33, Ireland's record cap-holder and goalscorer, said in a statement: "I want to acknowledge the tremendous job that Giovanni Trapattoni did for the past five years for the Republic of Ireland national team.
"In my view he has been a credit to the country and has carried himself with dignity as befitting a man who has given so much to world football.
"It is always sad to see people go but we are all aware it is a part of professional sport. I personally want to thank him for what he has done and wish him well."
Team-mate James McCarthy has expressed his surprise and sadness at Trapattoni's departure.
Midfielder McCarthy has earned 18 caps since being brought into the national side by the former Italy boss in 2010.
The 22-year-old said: "I'm gutted to be honest. He was a good manager and I enjoyed working under him.
"It is a blow to all the players. It has just happened - I didn't expect it to happen.
"We were in contention to qualify for the (2014) World Cup but unfortunately we slipped up on Friday and last night we conceded a late, late goal. It is disappointing, the players enjoyed working under him."
The start of McCarthy's international career was overshadowed by rumours he could switch his allegiance to Scotland, the country of his birth.
After making his first appearance in a friendly against Brazil in March 2010, it was not until he played a competitive fixture the following year that the speculation ended.
McCarthy feels he owes much to Trapattoni for bringing him into the fold.
He said: "It took me a while to get into the team but thankfully I got in and he stuck by me. He gave me my chance. He has been good for me."
McCarthy was speaking as he was officially unveiled as an Everton player at a press conference on Merseyside.
The former Hamilton midfielder joined the Toffees in a deadline-day move from Wigan last week - a transfer that may not have happened without Trapattoni's co-operation.
With the transfer deadline approaching on Monday, Trapattoni gave McCarthy permission to report for international duty a day late.
McCarthy said: "I'm thankful that he allowed that to happen. A lot of managers would have said, 'We want you in, fly over.'
"But with the time, at such short notice, it (the move) may not have happened - so a big thank you to him."
Defender Richard Keogh will forever be grateful to Trapattoni for handing him his international debut.
Keogh said: "I have a lot to thank him for because he gave me my debut for Ireland and for that I owe him a lot, really. I thank him for that.
"I think if you look at his record he has had a good spell and he has been great for the country and he leaves with his head held high."
Despite that, Keogh does feel the time may be right for a fresh approach.
The 27-year-old, overlooked for the latest qualifiers, said: "The last two games - losing at home to Sweden and then in Austria - have not been great and maybe it is time for a change.
"With a new manager coming in now it is a clean slate for everyone. I have heard a few names like Martin O'Neill and Mick McCarthy and if someone like that got the job it would be great because they are both good managers."