Professional bass player since 1986. Degree in music in 1992. Music teacher since 1997. In my experience through a huge variety of music, it's very rare and few and far between when you will find yourself strumming on a bass. I wonder if the wrong term is being used here. Are you actually strumming across multiple strings with one stroke of your hand/wrist or are you learning with a pick and trying to master picking up and down strokes on one string (alternate picking)? I can't for the life of me think of how the bass line to Fun Fun Fun should be strummed and will have to go back and listen to it just to check my memory.
As far as how to practice, I break it down into thirds for my students (and even myself sometimes!). 1/3 of your time on technique, 1/3 of your time preparing for your lesson & 1/3 of your time just playing what you love and want to play.
That being said, for the first two years I played, I didn't think about "technique" at all. I turned on my cassette player or radio and played along with everything I heard. Many great players have learned by doing this and nothing else. Practicing some technique (which I did endlessly later on) will help you to develop good physical habits, control and discipline as well as avoid some bad habits which could develop and be hard to undo later.
Bottom Line: Play! Find albums/songs/downloads/whatever that you enjoy and put them on auto-repeat. Based on your user name and the styles of music you mentioned, I highly recommend finding a Chuck Berry Greatest Hits album or two (I use his 2 Disc Anthology to teach from regularly) and listening to the variations on the same theme that were put in to many of Chuck's records. Add a bit of Jerry Lee Lewis, Little Richard, Bill Haley, Ike & Tina Turner and you're going to have a master class on how to play rock & roll bass. Huey Lewis & The News also had a cover album (Four Chords & Several Years Ago) which has some good examples of rock & roll bass playing.
Here's
wishing you the best!
Peace,
Mr DJA