Robinson on Monday signed a three-year deal at Cherries' League One rivals Huddersfield Town in a move said to be worth in the region of £250,000 to the Dorset club.
The 25-year-old follows Jason Pearce, Rhoys Wiggins and Danny Hollands (free) out of the Seward Stadium exit door this summer, transfers which have ploughed around £750,000 into Robinson's former club.
And the former Weymouth man, brought in by Eddie Howe in 2009 for a nominal fee, pleaded with Mitchell to ensure Cherries do not find themselves in the situation they were in when he signed from the Terras three years ago.
Robinson told the Echo: I think it is important that, with the money they have received from this summer's transfers, they try and give Bournemouth more of a chance to compete with the top teams in League One.
When I first joined Bournemouth, it was turmoil and the club can never go through that again. We were playing for our livelihoods and the club's existence.
Bournemouth should never be in that predicament again.
For me, the prospect of Championship football was too good to turn down, because, in my eyes, Huddersfield are the strongest team in the league.
They have a great fan base and are favourites to go up and if I can win promotion for the second time in my career then I'll be very happy. That's what I'm looking to do.
I desperately wanted to achieve that with Bournemouth because I love the club dearly, but it wasn't to be.
Robinson made 104 league appearances for Cherries, scoring 10 goals, and was part of the 2008-09 Greatest Escape squad before tasting promotion under Howe a year later.
The star, whose goal at Chester in April 2009 went a long way to securing Cherries' League Two safety following a 17-point deduction, added: It's been a fantastic three years and I have enjoyed representing the club.
I'm sorry to leave the boys, but when these offers are on the table you have to think about yourself, your career and your family. It was always going to be difficult to leave Bournemouth.
What we had at Bournemouth was very special and all the lads are very close. They are my friends and it was always going to be difficult leaving that behind.
I'd like to thank the fans too. They are a wonderful group and always supported me.