It will be probably be possible to gear up for raids only by drops, but you'll make it much harder (and slower) on yourself.
Allow me to explain:
While running heroics to gear up, you'll end up with a shedload of (triumph) badges anyway - every boss drops one on heroic. For about 120 of these badges you can buy an (unupgraded) 4-piece Tier-9 kit. That will provide you with a well itemized set and excellent setbonuses. There should also be 1 or 2 crafted items that are of similar quality, but these will require a few items (Crusader orbs probably, or runed orbs) which you can
also buy for badges.
While getting these 6 items together, you'll also have gathered a decent amount of the higher level (frost) badges, since the first random dungeon (by using the dungeon tool) gives you two of these for free.
By this time these should allow you to buy your 7th item already.
...And then you're almost kitted out already. I don't particularly like the badge system either, but dayum, it's fast to get geared up nowadays by using it.
My advice - run with it. Badgers will rule the world! (of warcraft)