Are there cards that let me cast cards from an opponent's exile pile, without the cards having to be exiled by another specific card? For example, Hedonist's Trove wouldn't be applicable here.
In a roundabout way you can use Cryptic Cruiser, Mind Raker, Murk Strider, Oblivion Sower and other Eldrazi to do things with an exiled card an opponent owns from which you can use a variety of theft cards to gain control of said cards. You'll pay a hefty price for that kind of atrocity though.
What cards do you recommend for stealing from an opponent's graveyard?
Depends... is this an EDH thing? Casual? Legacy? It helps to know what the parameters are.
I'm going to assume EDH and are the colors.
Well... there is, in no particular order:
Not recommended long anything specific. That's what I found by doing a search. As you can see, some are expensive.