I also store my cards in 3 ring binders and lay them flat. The only problem I've encountered with this is that the plastic pages sag down from the round rings until they come to rest flat against the inner back cover. Binders with "D" rings, however, alleviate much of this sagging problem.
Then there are binders whose rings are mounted on the back cover of the binder. This makes it easier to load paper into the binder, plus the pages lie nearly flat. With center mounted rings, the pages will never lie completely flat.
I've also found that thinner binders with "packed" contents work better. The tighter the pages, the more mass, hence the greater the force required to disturb anything - lying down or standing up. Wide binders with a only few pages inside allow the pages to sag and flip-flop this way and that.
There are also "memory" boxes (sold at AC Moore, in the wedding stationery section of Staples, etc..) I use these to store the WP billboards, etc. - basically those items that binders cannot reasonably accommodate.