Richardson will move into the role of assistant manager at the Crown Ground when Paul Cook takes up the reins on Wednesday, and he was disappointed to end his temporary tenure with a loss.
Shaun Harrad scored the only goal of the game from the penalty spot after Bryan Hughes brought down Mark Bradley in the first half.
“The lads will be the first to hold their hands up,” he said. “We didn’t get beat by Rotherham, we beat ourselves.
“The two halves were completely different. We didn’t do the things we are good at, we didn’t get on the ball in the right areas and made wrong decisions at times. But as long as we know we are not playing well and the reasons for that it’s not a problem.
“We move on and make it right and we will win more than we lose. We have had a long chat in the dressing room, we are still positive and we still have the same goals and if we do the right things in the right way we will be up there and finish where we want to be.”
Middlesbrough climbed into the Sky Bet Championship play-off positions as Patrick Bamford's first career hat-trick secured a comfortable 3-0 win over Leeds.
Louis Longridge and Greg Docherty will leave friendship at the side of the pitch for 90 minutes on Sunday when Falkirk face Rangers at Ibrox.