If I did the dumping, nothing. I dumped him for a good reason.
If he did the dumping, if he got on his knees and begged for me to take him back, I might be willing to discuss it. 
I also think another factor that goes into it is how long the couple has been broken up before one begins to suggest that they get back together.
For example, a guy I was going out with in college dumped me the year after I graduated. He was a year younger and had just graduated himself, so was planning on going back to where he was from. Before he dumped me, we had plans that I would get my PhD near where he lived and we would eventually get married and have kids, etc. When he dumped me, he said he did so because he wasn’t sure if I was THE one and wanted to find out before he saddled himself with the same person forever. We got back together after being broken up for two weeks, then finally he dumped me permanently and went back to where he was from. After agonizing over the breakup and feeling horribly sorry for myself for about six months, I realized what a turd he was. About six months after I had that startling epiphany, he started e-mailing me asking me if I was still interested in living with him, wondering if I was dating anyone, you know, fishing. At the time, I was going out with the man who would later be my husband. There were several reasons I would never go back to him. First and foremost, I don’t cheat. Second, he dumped me - twice - and I’d be investing a lot in the possibility that he wouldn’t dump me a third time.
Third, and perhaps more important to this particular subject, after he dumped me, I saw things more clearly. This guy was an asshole. He got upset at me for stupid things, like eating more than five grams of fat at a time (he was worried I’d get cancer) and not knowing classical composers (he actually quizzed me on it, and I let him), and other dumb stuff. When I had the time to think about who he was, I didn’t like what I saw, and my parents and friends had always hated him because they saw 20/20 when I didn’t and recognized him for the jerk he was. If it had been, perhaps, a month after he had dumped me, I might have considered getting back together with him, but after enough time, I could see pretty clearly that we were not meant to be together.
In my experience anyway, once a relationship has soured and ended, women tend to have a longer memory for what went wrong.
I’m with elmwood. I’ve heard all of the tales about how so-and-so was an SOB, but a few weeks later, he’s getting another at-bat. Feh.
You don’t have to date a girl to get “burned”… I had a female friend cry rivers on my shoulder about her ex… and how he got hitched way too fast to another b*tch… etc… Naturally we end up saying bad things about that ex of hers…
Needless to say… they got back together less than a month later… and I felt like a complete idiot. Eventually they married and moved out of town… thank god.