Do you have the training and gear for traveling on steep snow?
In addition to Mt. Hood, Mt. Jefferson and the Three Sisters, the other section that may give you problems that early is the Crater Lake/Mt. Thielsen area (the latter being the highest point on the PCT in Oregon). Here is a way to keep track of snow depth:
Oregon snotel sites: http://www.or.nrcs.usda.gov/snow/maps/oregon_sitemap.html
Note, though, that most of the snotels are at lower elevation than the PCT, and there are none in the high elevation wilderness areas. Also, if you check the comparison graph, it uses SWE (snow water equivalent), so expect the snow depth (in late spring, when it becomes soggy) to be about double that.
If you can postpone your trip a couple of weeks, you will have a much better chance! The average meltout for the highest sections is generally mid-July.