The Holland striker has joined Turkish giants Fenerbahce on a three-year deal, ending his six-year stay on Merseyside.
The 31-year-old found first-team starts limited under Kenny Dalglish last season, and even the Scot's departure could not stop him leaving.
"I think it's great for the club that they have finally announced the new manager," Kuyt told Liverpool's official website.
"From what I have heard from (Swansea goalkeeper) Michel Vorm, who is with me in the national team, he is a great manager and he told me he will be perfect for Liverpool and he will definitely lift the club up to a higher state."
Kuyt insisted the appointment of former Swansea boss Rodgers on Friday was not a factor in his decision to leave.
"I would also like to say my decision to leave Liverpool was nothing to do with the new manager coming in whatsoever," he added.
"I just had the feeling that I had to go somewhere else for a new experience after six years. I made the decision but it's not about the new manager. I have great memories from my experience with the club.
"I'm sure Liverpool will improve under the new manager. Liverpool has a great squad of players and I'm sure with this manager they will do well. From the bottom of my heart I wish them all the best, wish the new manager all the best and I hope they will be very successful."
Kuyt, who had a year to run on his Anfield contract, had been expected to leave Merseyside this summer after a season in which he scored just two Premier League goals.
Fenerbahce offered the prospect of Champions League football next season, something not available at Anfield.
The Istanbul giants will enter the preliminary rounds of the competition after finishing runners-up to Galatasaray in the Super Lig.
Kuyt is preparing for Euro 2012 with Holland and a delegation from the Turkish club travelled to Amsterdam to seal the deal.
The Dutchman said: "It's a great new experience. I had interest from several clubs in Europe and once Fenerbahce came in for me, with the ambition they have as a club, and also by showing how much they wanted me, for me it was an easy decision to make."
Kuyt leaves Liverpool having won just one trophy in six seasons, the Carling Cup last season, when he scored in the penalty shoot-out win over Cardiff.
The Dutchman scored 71 goals in 285 games for the Reds.
He was never as prolific at Anfield as he was during his time in his homeland, although he was often played out wide.
Neymar's move from Santos to Barcelona is no doubt the biggest summer transfer.
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