Non-ionizing radiation, like radiation from mobile phones, electric power transmission, and radio frequencies, etc. is regarded by the majority of people as safe to the human body. However the World Health Organization in 2011 stated:
The WHO/International Agency for Research on Cancer (IARC) has classified radiofrequency electromagnetic fields as possibly carcinogenic to humans (Group 2B), based on an increased risk for glioma, a malignant type of brain cancer, associated with wireless phone use.
This release is not definite, just a possible link, and it has been refuted. I cannot seem to find any good study that shows a direct association between health issues and non-ionizing radiation.
There is this study that showed brain glucose metabolism linked to cell phone proximity, but it has been refuted because it only involved 47 people and didn't control for the effect of the heat generated by the phone. And similar issues with other studies.
So my question is, is there any scientifically accepted link between non-ionizing radiation and health issues?
Your question is very broad - you've asked about non-ionizing radiation generally, rather than say specifically about radio waves. And that makes it much easier to answer in the affirmative.
Also if anyone cares, I asked this question because of a discussion that took place in the comments of this answer on SE's Electronic site, and I was surprised it hadn't been asked/answered here (there are a lot of somewhat related questions, but I couldn't find what I wanted.)