I use Armor All Multi-Purpose Cleaner (not the more popular Protectant) to get the dirt out of interior plastics. It'll even work on plastics inside the engine bay.

For stubborn spots, I spray and leave it on for about a minute to soak, then wipe off with either an old rag or microfiber towel. Works well on door sills and window pillars that tend to sprout water stains with white edge residue.

The more familiar Armor All Protectant can be used after cleaning is completed to impart the shiny finish.