There are quite a lot Bomberman games. Though, there doesnt seem to be a strict Bomberman-Franchise, and seems like its comparably unknown. Well, in many games I have heared the appeareance of a Theme. Investigating for the origin of that theme, well, seems like this was the main theme of the original Bomberman game. There it is:
In Pocket Bomberman, a game for the Game Boy, which is a sidescroller (not a „classical“ Bomberman game), there is an enhanced version of this. Unfortunately, I couldnt find a video, which only contents this music, so here is one with the gameplay of that game – the music starts at 0:12.
I also found a video claiming to show „Super Bomberman 3“. Whatever game it is, it seems to origin from the Bomberman Franchise and has a lot of versions of this theme:
As always, all videos are not made by me, just found on youtube – so, well, click on them, comment and rate them.