I'm trying to get up there again this year so I've been watching this also. I'm hoping snow conditions will hold until mid-June or I find another partner that can go earlier. We plan to do the east side so it's a little longer season than Avy Gulch, which gets hit hard with afternoon sun. Conditions are great right now, but they will deteriorate rapidly in a matter of weeks. By July, it will be probably be crappy, but as others have written, it can be done any time.

Fifth Season has good beta - usually on a recording and here's another link by the Climbing Ranger:

Mt Shasta Climbing Advisory

Best time to climb: The best time to climb Mt. Shasta is generally from late May to mid July on the SW side of the mountain when summer days are longer and the weather is generally stable. However, in dry years (2007/2008), the thin snow pack creates the best climbing conditions in April, May and early June. When the snow melts away, you are left with 7000 feet of scree, talus and boulders. In heavy snow years the climbing season extends to August or September. There is NO trail to the summit. Climbing is much more safe and fun on consolidated snow!

Edit: Kevin snuck in a post with this same info while I was typing. One last thing, last year in June there was still snow, but the sun cups were huge, which made for difficult footing. And it was really icy going through the Red Banks chutes. So even if there's still snow on the route, late season can be poor conditions. Still a great climb.