Operation Periodical Discipline requires a multi-faceted plan of attack. The primary front plays out on the Internet: get as much of your info as you can online and cancel corresponding subscriptions.
The secondary front is interpersonal: enforce a household rule that if a publication is more than three months old, it must be given away or recycled, even if it's still unread (or even unflipped-through).
And if the enemy still broaches the line, you'll have to send in reinforcements: a nice magazine rack (or two, at most) with enough capacity for no more than three months of glossy softbound reading material. Limit what you will take in to to what your rack can hold.
We nominate this one, from Furni. It comes in veneered beech, dark or light, with a white, bent acrylic pocket and rubber feet, for $88. Furni is also selling their samples right now however, and these are a better deal, at $66 apiece.