I think it would be great. Just the announcement, and record it as a post game stat. Show where assists occurred for a champion as well in match history. There's a huge difference between someone who gets a lot of assists early game (and hence was likely a huge factor in earning the kill) and getting a lot of assists late game (when your team is stomping you'll rack up assists even with poor play). It would be nice to be able to differentiate.
Back on topic, assist sprees could just be announced at higher lengths to prevent announcement spamming, and could be only a visual icon if other announcements are in queue to be made. I'd hate to hear "adc killing spree" "support assist spree" go off every time bot Lane gets a kill, but it would be nice to hear in most instances.