A salad chopper I can put in the dishwasher?
A lot of formally not dishwasher-proof gadgets - if they are not electric, and do not contain wood or non-stainless iron parts - do fine in the dishwasher. The most critical thing here would be the blades, and at least one of the ads suggest the blades are dishwasher proof - no sharp blade LOVES being in a dishwasher, but this suggests the alloy chosen being corrosion-proof enough to get by.

Alternative or "green" detergents, or minimizing detergent use, might be a bad choice though, since these are often less effective at dissolving/dislodging food scraps inside things.