I always say, despite the huge amount of trouble it gets me into, if you want to get stuff cleaner, get a woman to do it, if you want to get it clean, do it yourself.
Bottom line is whether you want to take it to pieces or not. Soda (or vapour if you're spending money) blasting is best if you're stripping everything apart.
If you're not, Scotchbrite works best when it actually says Scotchbrite on the packaging, and when it's not sold as a kitchen scourer. Most engineering or autobody paint supply places sell it in several different grades.
Either way, you want to think about what's going to stop it oxidising again.