Hello Richard,
This is a yes and no answer.
Yes, listening to the Liturgy of the Hours is a perfectly valid way to pray it. If you are actually praying along with them in your head and focusing on your prayers, your body can be doing repetitive work. The minor hours of the Liturgy of the Hours were sometimes prayed this way in monasteries in Medieval times. If they were out in the fields working, usually they would stop and pray the hour in the field, but there were also times that they sang the hour as they worked. If my brothers are driving for long periods of time, many times they will put on the Liturgy of the Hours to pray as they drive.
Here’s the no to the answer. If you are listening to the Liturgy of the Hours and your mind is not engaged in the prayers, but the work in front of you (like doing your taxes) it would not be a legitimate form of praying the hour. You can still listen to it as background music/noise, but for those of us who are obliged to pray the Liturgy of the Hours, it does not fulfill the requirement.
Hope this helps.
God Bless,
Br. Ben, CRM