It looks like Flynn was fired or forced out by the Defense Department. Whether that was for cause or for expediency is anybody's guess. "The Los Angeles Times" is nobody's idea of a right leaning publication or an anti-Obama publication and their 2014 account of Flynn's leaving the Defense Dept. seems pretty straightforward that he was trying to reform and modernize a dinosaur that didn't want to be reformed at the Defense Intelligence Agency. The article was updated to indicate he was well thought of at the DIA.
Lt. Gen. Michael Flynn quits as chief of Pentagon intelligence agency - LA Times
Flynn's security clearance was renewed just last year though so the Defense Dept. obviously did not do any vetting to renew that. And that would suggest Flynn had a clean record so far as the Obama Administration was concerned--this would be my guess. But if Flynn didn't divulge improper payments from Russia, I don't really blame the Obama Administration for not catching the problem at that time but it did happen on their watch. So far Flynn has not been indicted for any crime but there is only speculation that he may have violated U.S. law.
AND I don't fault the Trump Administration for taking him on based on his government credentials and reputation and high level security clearance. And I don't fault the Trump Administration for letting him go when he misled them. None of any dealings with Russia occurred since the election, however, and there was nothing to indicate it was a problem at the time.
But saying that it was on Obama's watch or by Obama's White House or the Obama Administration is not saying that it is personally Obama's fault. But it did happen on Obama's watch and the Trump Administration had nothing to do with it. And the buck stops here as the saying goes.
The media is still hugely hypocritical in nitpicking semantics regarding anything in the Trump administration and using that method to deflect from the actual issue that they should be focused on.