I was playing this game in-person with someone and I was white. My opponent had a pawn on a7 and kept moving his king back and forth between c6 and d6 at least 3-4 times while I maneuvered my bishop around the board to capture his pawn on a7 to make room for my past pawn. After I captured his pawn on a7 he moved his king back to d6 and claimed it was a draw because at that time he had moved his king back and forth on the same 2 squares 6 times. I understand the 3 fold repetition but his king had other squares to move and wasn't in a perpetual check. Was this a valid draw?
No, it was not a draw. For a draw by three-fold repetition, the exact position, and all of the possible moves (castling and en passant, which since en passant can only happen on the move, in essence, it requires a fourth repetition of the position), have to be repeated three times.
It was clearly a case of your opponent not really knowing the rules.
It seems like he was somewhat confusing, and mixing, three-fold repetition with the 50-move rule, which requires 50 moves of just moving around without a pawn move or capture with repeating the position three times.
P.S. The repetitions do not need to be consecutive.