Dishwasher Powders - Nope! These contain abrasives that will scratch up your bricks.
Liquid Dish Soap - Probably anything goes here. Lemon scented bricks, Mmmmmmm!
Head soap - Most anything here, but avoid stuff like liquid lava w/ pumice.
Laundry Soap - Seems to work fine, I have filled entire wading pools with water, Lego, and laundry soap to clean vast quantities all at once. This is your economy of scale if you have to wash a lot efficiently.
Bleach & Water - Not a great solution, unless you are just briefly dunking it. Bleach will clean very well, but if you leave it soaking for hours or days, Lego slowly absorbs the bleach and swells very slightly causing the bricks to gain clutch power. Not a bad plan if you want to put some stick in your old bricks, but experiment carefully before you commit. This swelling doesn't seem to cause any long term damage.
H2o2 + Oxyclean - This will 'lighten' UV damaged bricks, but I haven't had much success with it. It takes a long soak to accomplish it, though. There are many pages on the Intarwebz talking about how this works.
Petrochemicals - These are in general risky, as they can act a solvents for ABS. Test before use. Except for....
Goo-Be-Gone - I have use this on all types of Lego, and it is perfectly safe to drench on. Great for getting rid of sticker goo.
Warm water is fine, but water > 130F starts to get risky, aside from burning yourself, thinner types of plastics start to bend and warp. Watch out for things like base-plates which are particularly vulnerable here.
Running Lego through a dishwasher is probably not a good idea, they generally use pretty hot water to kill germs, and the heat exposure isn't brief.
Also, be aware that Lego isn't one type of plastic. It is mostly ABS, but there are other things like windshields mixed in, along with differing formulas of ABS. Heat and chemical tolerances will vary.