Pochettino left St Mary's to take the vacant post at Tottenham, where he was handed a five-year-contract at the end of May.
The Argentine guided Saints to a club-record equalling eighth-placed finish in the Barclays Premier League last season and Rodriguez was full of praise for his former boss.
"Mauricio is a great manager and a great guy and was well-respected by the lads," Rodriguez said.
"He worked us really hard and we got the good results and finished eighth which we were happy with.
"Football moves on though so it's one of those things."
Striker Rickie Lambert followed Pochettino out the exit door to join Liverpool earlier this month and there are fears more players could do the same.
"You don't want to see players going and the team being taken apart," said Rodriguez.
"You can sometimes never stop it, it is one of those things, so I'm trying not to think about it and hopefully we will have everyone at pre-season."
Jay Rodriguez talks about England's chances at the World Cup