Why is an H bridge needed in a bidirectional motor control circuit?
If so, I do not understand the comment that says the diodes must be
  able to pass the full current of the motor, because it seems they
  would only be used when the motor is stopped, which would produce only
  a transient current through the H bridge, and not pass current while
  the motor is being run.


The H bridge supplies the motor with current and the instant that one side of the H bridge is controlled to go open circuit, the same motor current will still flow and that current will cause a large back-emf if there isn't a low impedance path for it to naturally decay to zero.

The diodes therefore have to be able handle full load motor current for that brief period of time when the current has to decay to zero. For PWM control this is quite often every cycle of PWM. For plain ordinary on/off control, the current is only going to flow through the diodes when the motor is turned off.

In summary, current doesn't instantly become a lower value when the motor is open circuited.