Ok — yes, I know pipe cleaners may not be a common household item, but you can get them at any craft store and they're a handy thing to have lying around.
Using a vacuum or hair dryer, blow out any dust that's worked its way deep down into those crevices. Then lightly spray a pipe cleaner with a disinfectant and use it to scrub the inside of each slot. Finally, grab a dry pipe cleaner to go over the slots one more time to absorb any excess spray and moisture. Voila, your knife block is sparkly clean!