What I did to evolve my Azurill quick was go to the area I caught it(Bulbapedia says you get bonus happiness in the place you met the Pokemon) and just ran back and forth for like an hour. I just watched tv while doing this.
Well, I caught these in a cave and Cleanse Tag seems to reduce encounter change by like 1%, so that's out. I guess I can just bike back and forth and hatch some random eggs while I do homework. I think I can write a lab report one-handed. If that is true it should make subsequent generations slightly easier since I'll have met them on Egg Hatch Road, but regardless this is a HUGE setback. This adds many, many hours to the process of making a team. I don't know how many yet, but a ton.
Edit: Wow, Luxury Ball actually has a use now. I may have to throw out this batch of Chinglings, buy a bunch of Luxury Balls, and catch a new batch. Apparently the kind of ball something is in is now an inheritable characteristic (Pokesex is super weird I guess?), so I assume the Luxury Ball effect is also inherited.