Magnets, Velcro and binder clips are all helpful in managing cables. But they're all probably less abundant in your house than tape. The only problem with tape is the sticky residue, but this folding trick can eliminate that annoyance.
As Reddit user wetneon shows, you can take a length of duct tape, cut out a portion to remain sticky on each end (not unlike those paper wristbands you get at events and clubs), and fold the rest over onto itself. You're left with a strong, durable wrap with the holding power of duct tape but none of the mess. It's a relatively simple way to get a custom-sized wrap for your cords.