In June, there have already been several thunderstorms and rain showers. But, a sustained monsoonal flow has been mostly absent. Dew point temperatures will rise throughout the week, with most Arizona being in the mid-40s by next weekend.
While isolated thunderstorms would be come more frequent this week, next saturday is the first day that seems to have pretty good areal coverage.
One last note, the start of the season should be strong. The 6-10 and 8-14 day outlooks from the Climate Prediction Center show above average precipitation. The longer range 1 and 3 month outlooks only show normal precipitation chances. I think these may change the next time they are run since the El Niño outlook has a decreased likelihood of it occurring this year.