Ethiopia gave the birth to Ethio-jazz (Ethiopian Jazz or African Jazz) popularized by Mulatu Astatke and his band.
You will possibly have difficult time looking for one of the true Ethio-jazz live performances (as the authors of this blog post evidenced) as the most live performances are a mix of contemporary Amharic music, smooth jazz, pop or even reggae with little of Ethio-jazz to be heard.
We recommend Ghion Hotel night club (near Meskel square), where live performances are part of African Jazz Night every day except for Monday (free entry).
You can also try Jazzamba lounge in the Addis-oldest Taitu hotel, but you will likely be dissapointed by the music as well food and cover (80 Birr).
Live bands playing contemporary Amharic music (pop etc.) can be found almost anywhere, just ask the locals. Expect the synth keyboards as a background for mostly average singers.