Just adding my . 02... Armor all. Original. If I don't want shiny, I spray it on, use a tire brush to clean, and then spray it off. Usually, when I'm going to wash the truck, I spray the whole truck, and then do the tires first. By the time I'm ready to rinse, the armor all has done it's job, and I just spray it off. Clean, but not shiny.
If you do want the shine, ( I know you said you don't want shiny, but I'll put this so you know what not to do
) Armor All tire foam works, and it usually takes two coats. Just spray it on, and when it sets for a while, you'll have clean, semi-glossy tires. One more coat, and they'll be shiny.
SOLER
If you do want the shine, ( I know you said you don't want shiny, but I'll put this so you know what not to do

SOLER