"When I was younger, our catechesis consisted mostly of memorizing questions and answers. It seems like we do less of that now, and I worry that our students don't know their faith. Does memorization have a place in catechesis?"
It's difficult to live a faith we don't know, and yet it's also true that simply knowing something doesn't guarantee that we will live it.
For this reason, some memorization might be necessary for formation in the faith, but it is not sufficient. We also need to be conscious of the fact that educational methodology in general has changed a great deal from the time when rote memory was the norm. Many students aren't used to learning things this way, and others might have particular difficulty with memorization due to learning problems.
When teaching something you want the children to memorize, consider using music and/or rhythm, which can make memorizing things dramatically easier. Introduce new terms and definitions with hidden words or letter guessing games to make these activities engaging and fun. Finally, remember that it's understanding and desire, not memorization of prayers or definitions, that indicate readiness to celebrate sacraments. It would be contrary to Canon Law to withhold sacraments because a child has trouble with memorization.