Why memorize?
The whole idea of memorizing during the elementary years is to take advantage of a time in life when it is relatively easy to memorize and fill a child's mind with useful information that act as "pegs" from which future learning can be hung. They do not need to fully understand it at this point because that will come later, in the dialectic years. We know this is true of helping children memorize Scripture verses, as we drill John 3:16 into their minds and hearts before they understand salvation. It is also true of academic information.
What we memorize
Bible verses. We want our children to have God's word hidden in their hearts. One way we have been doing that this year is to enroll them in an AWANA program where the emphasis is on children memorizing Bible verses in a fun and competitive way. Even our four-year-old with speech delays has enjoyed it and memorized a verse on his first night there. Our eight-year-old daughter is also part of American Heritage Girls. Her troop verse is Philippians 1:1-16 and every girl who memorizes the whole passage earns a special badge. Guess what we will be working on next year?
Some other resources we enjoy using at home for this are:

General Bible knowledge. I shared this in my last article about the 7 Rs, but it is worth saying again. I am so grateful that the leaders of Classical Conversations thought up a Bible component of the Foundations memory work. It is not part of the regular Foundations (ages 4-11) memory work, so we work on it on our own. The best part for all of you is that it is available to everyone for free!
Catechism: We attend a Baptist church, so this is not part of our faith tradition, but I like it. Catechism is nothing more than asking questions of children that they memorize the answers to, so that they can learn the fundamentals of their faith. One free resource I use is an app from New City Catechism. It contains questions and answers for both adults and children, as well as background information, prayers, and Scriptures that go along with the topics. Some of their songs are from a ministry called Songs for Saplings, which takes biblical truths and puts them to music for children to learn. Another book resource, again from Susan Hunt, is Big Truths for Little Kids: Teaching Your Children to Live For God.
Bible timeline: Finally, we want our children to know the timeline of the Bible, to recognize what happened when to whom and where, and one of the best ways I have found to do that is through Grapevine Bible studies, which helps children draw a timeline of the Scriptures, from Genesis to Revelation.
The CC apps. They are pricey as apps go, but worth it in my opinion. It is the same information as the online tutorials on the Classical Conversations website, but you do not need to be online to access it. I can take my iPad with me in the car and hand it to my daughter in the back seat and tell her to review her week 12 memory work and it is all right there.
CC Connected. This is not just one resource, but a collection of resources contributed by other CC parents. We have found many memory helps here, especially flip books that have been created for each cycle, and worksheets to do at home to help memorize the work.
Copywork. When we have not had a specific worksheet, good old-fashioned copywork has done the trick, too. We have used this kind of journal for several years, where my daughter can copy her work and draw a picture to go with it. There is also a great website that will generate handwriting practice worksheets for you.
If all of this sounds like a lot, let me assure you that we don't do it all everyday. We vary Bible verse memorization with Bible information memorization, and we might do catechism one week and Grapevine the next. We don't even do all of our CC memory work at once, but sprinkle it throughout our school day and change it up between the app, worksheets, and just listening to the CC audio CDs.
What are your thoughts about memorization? If this is something you focus on (whether you homeschool or not), what works for your family?