I third the Luna reccomendation...You can also try Carmen's De La Calle Cafe on Thursdays for some great live jazz, and the new Kirby's steak house up on 1604 has been booking some good acts. There's also Dolores Del Rio on the riverwalk and Stephania's in Monte Vista on Saturdays and Sundays.
If you want to stay in the know, listen to San Antonio's jazz Station - KRTU Jazz 91.7 (I work there)...
krtu 91.7 fm | jazz for san antonio.
We bring in the legends of jazz twice a year...over the past five we've brought Dave Brubeck, Herbie Hancock, Poncho Sanchez, John Scofeild, McCoy Tyner, Chick Corea and others...on March 28th, it's DIANNE REEVES. Tickets are on sale right now. San Antonio is a great jazz town and it's only getting better, welcome!