There are many synonyms for the phrase "play jazz," and each one brings a slightly different nuance to the meaning. One alternative is to "perform in the jazz idiom," which suggests a certain level of expertise and familiarity with the genre. Another option is to "improvise in the jazz style," which emphasizes the creative, spontaneous nature of jazz music. Alternatively, you might say someone is "jamming" or "grooving" in a jazz context, which connotes a sense of energy and enthusiasm. Finally, if you want to emphasize the social aspect of playing jazz, you could say someone is "swinging" with a group of musicians.