My girlfriend and I hiked part of the North Rim starting on June 19, finishing the 21. We went up the Rockslides Trail and down Yosemite Falls.
There was still a lot of snow around the Ribbon Meadow area. It was consolidated, so we didn't posthole much. However, navigation was a little tricky because the red markers on the trees were spaced out quite a bit. We crossed Ribbon Creek a couple hundred yards upstream of the real crossing on a combination log/snowbridge. I am sure by now that is unusable. However, the creek didn't appear to be all that fast, I am sure a spot can be found to cross through safely. Eagle Peak Meadows was basically a swamp or walking on snow. Expect to get your feet wet and have them stay wet the whole day.
We decided to go down the Yosemite Falls trail instead of Snow Creek because we both were a little pressed on time to get back to work, and decided the inevitable walking on snow most of the day around the North Dome area would be too much. So we went down a day early and played tourist around the valley.
Here are some pictures of the trip:
https://picasaweb.google.com/braap13/NorthRimYosemite