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 ...
The official guidance from The LEGO group suggests using a mild detergent.
Cleaning your LEGO® bricks is really easy! We recommend that you clean
your LEGO® parts by hand using water no hotter than 104°F / 40°C and a
soft cloth or sponge.** Higher temperatures may affect the quality of
the parts. You can add a mild detergent to the water - please
rinse them ...