Can I Run a Background Check On Myself?

I am from Michigan, am now live in Oregon. In 2006, I had some felonies which were dropped to misdemeanors (in Michigan).

However, there was a point in 2007 when an employer did a background check and I was dismissed because the felonies were viewable on my record. I’m 100% sure they were dropped to misdemeanors — according to the HR for that company, the background check showed the felonies AND misdemeanors as a list of charges, not discerning between what they were originally and what they were dropped to. I tried to get to the bottom of it back then, but it seemed like everyone I talked to at the courthouse and police station in Michigan had conflicting information. Confused and frustrated, I never really got to the bottom of it.

So now I’m looking for work in Oregon, and I’m very curious as to what potential employers see when they run a background check on me.

How does the background check system works and is there a way for me to run a comprehensive one on myself?