The 74-year-old, who took charge of the side in 2008, left his post by mutual consent on Wednesday morning.
Keane 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."
Trapattoni has expressed his desire to stay in football management.