This might be overkill, but I just bought a used '02 ES 300 and I wanted the mats as clean as I could get them. Vacuuming and extraction are just not enough to get down into the fibers.
Here's a tip on cleaning extremely dirty, winter stained carpet mats: It goes to an extreme but will remove 100% of dirt and dust from deep in the carpet fibers. My mats were cleaned by a detailer twice with their extraction machine and it didn't do a thing to improve the mats appearance because it only extracts surface dirt.
If it's warm outside: hit the mats with a hard spray and the soak the mats in a mild cleaning solution , fibers facing down, in a large shallow tub or a large plastic storage bin. Dirt will release downwords.
If it's cold outside: use your indoor work tub or your shower and hot water to clean deep into the carpet fibers. If you have a spray hose, use it like a power washer with hot water to get deep in the fibers. Again soak the mats in a mild solution for a few hours, face down. Let chemical do its job.
The concept here is using water to remove 95% of the dirt and using a mild chemical with time to remove the grease or stains.
Final cleaning: Use a scrub brush and cleaning solution on the mats, rinse, and hang up in the sun to dry.
If there is any thing left after these cleaning and soaking steps, your mats are permanently stained and consider having them replaced.