What mechanics exist that trigger Burning Vengeance?
Non-flashback options include:


(Standard/Modern) Creatures you can cast from the graveyard, like Gravecrawler and Havengul Lich.
(Modern) Retrace cards like Raven's Crime, Worm Harvest, Flame Jab.
(Modern) Haakon, Stromgald Scourge, especially played with Tribal cards like Nameless Inversion -- this is a "real" Modern deck, although not "tier 1".
(Modern) Toshiro Umezawa, although really that's flashback by another name.
(Modern) Chandra Ablaze.


Overall I feel like it plays best in a "Jund"-style deck (basically a huge pile of two-for-ones, building up card advantage incrementally while pressuring the opponent to keep tempo). In particular, I could see using Burning Vengeance in place of Seismic Assault in an Aggro Loam variant that doesn't mind losing a little bit of speed for card advantage -- pitching a land to Raven's Crime and getting your Burning Vengeance trigger only cost {B} more than pitching it to Seismic Assault, but you get to force your opponent to discard.

I don't like trying to combo out in one turn with Burning Vengeance because I feel like it's weaker than Pyromancer's Swath in that kind of deck.