I wish I had a good suggestion for it, but I don't. 
If they overwrote each other, the announcer would stutter and it'd sound terrible.
I don't want them removed because I love the voice, it reminds me of the announcer yells in Blood 2 death-match a lot.
It might slightly cheapen the effect, but maybe only have the announcer sound off for the first kill and not be able to sound off again until his current sound is complete, and skip the ones in between? So, in a fast-paced bloodbath event, the announcer might only get to yell once for every 2-3 kills or so. The kills could be shown on screen still, maybe just make the text and champion icon a little smaller and let them scroll up like scrolling combat text in World of Warcraft?
You'd get something like:
KILL "OWNED!"
KILL still yelling owned" - doesn't get an announcement
KILL "GODLIKE!"
KILL still yelling godlike - doesn't get an announcement
KILL still yelling godlike - doesn't get an announcement
KILL "ANIHILATED"
etc...
It's not ideal, but it doesn't happen THAT often, and it can't continue too much longer since most everyone has died.