I've got a Blue Star stove with heavy cast iron grates and burner
bowls underneath. I cook a lot and inevitably get oil, sauces and
other gunk on the grates/bowls, which cooks on. I have tried running
them through the dishwasher but the oil seems cooked on.

Any suggestions on how to clean them up? I've read some folks use oven
cleaner in the normal way. I'm hesitant to put them in a
self-cleaning oven but that seems overly paranoid.

Or maybe I should just let them get gunked up, evenly glazed with oil
over time, and periodically wash the non-oils off in the dishwasher.
I bought the oven to cook on, not as a fashion statement. :-)

Thoughts?