Due to record snowpack and a cold wet spring, nearly all those high places in the western US will still be under many feet of the white stuff. The exceptions are Arizona and New Mexico, where the problem is avoiding the many fires that are unfortunately ravaging that drought-stricken area. (There are also extreme fire restrictions in that region, for obvious reasons, so you'd have to take no-cook foods.) All of us out here are watching snow depth and stream flow charts in hopes of seeing some signs of melting. Places normally melted out by early to mid-June are still solid snow.
Check out this site: http://www.nohrsc.noaa.gov/nsa/
In the top map, click the region you want. Then go down and click on the map for "Snow Depth."
You might want to consider the northeastern US (Adirondacks, White Mountains) or Michigan's Upper Peninsula--no snow issues there! The Superior Hiking Trail and Isle Royale are also good places. If they weren't so far from Oregon, I'd be tempted to go there!
In the west, Hell's Canyon is snow-free, although it's liable to get very hot if you go down into the canyon. I just saw a recent trip report here on BPL, and some trip reports from earlier this year are on portlandhikers.org.