All of the pulleys turn freely by hand and the belt is in the right places (in all pulley grooves and all in line), yet the mower will not run when the pedal is pushed. Sometimes when I rock the mower back and forth the belt catches and I'm able to move for a little bit, other wise nothing grabs.

Make sure it is on correctly then make sure you are lowering it down and if not it is either broken or the transmission is not connected to the mower

Have you had either one of the back wheels off ?