You'll most likely have snow on the Washboards and in the Grand Staircase. There are also some other pocketed areas where snow lingers. When I am on the East Face of Whitney early season I prefer a mountain boot versus climbing shoes. You don't get as much friction on the slabs but it makes the rest of the ascent easier. The boots also help with the descent down the Mountaineer's Route which will still have a lot of snow. Depending on the time of day and conditions you may need crampons to safely descend it.
Thank you for your information Kurt. I found some photos from 2010 and 2009, they shew that washboard was covered by snow at late May. We decide to add an alternative route, or say change the route to East Buttress. At least, there's not too much snow on there according to the webcam.
By the way, I saw photos of you on Sierradecent, climbing east face.