Would baskets instead of drawers work? That's my first thought, but if you're storing really small stuff I could see the appeal of the organizer you linked. If that's the case, maybe instead of drawers, organizer cases? Here's an example, and Plano in particular has a large selection of things like this. These would allow you to utilize the vertical space in the cabinets over baskets.

I know vintage wood library card/index card file boxes would work (and look amazing), but that can get pricey. 4&6 drawer ones go for $100 and up on Ebay.