Yeah, dish soap should keep the bearings clean!

I had a tube of nice White Lithium grease that I always preferred, but now that it has run out I've got a tub of auto parts store "wheel bearing grease". Not as pretty as the White Lithium, but I'm sure it works fine. It is a little more viscous.

I'm pretty sure the various grades of grease have more to do with heat rating and breakdown than what device the bearing might end up in. Bikes would be totally non-critical on that scale.

David