Take a baby wipe and wipe down the entire shoe, soles included. This will likely be extremely dirty. Use as many wipes as needed. Let the shoes air dry for a minute so you'll be able to see the really dirty spots and streaks.
Gently rub the spot in a circular motion with a clean baby wipe. The spot or streak should come off with continued rubbing. If it doesn't, use a Mr. Clean Magic Eraser Pen on the spot. Dab the spot with the end and rub in. The spot literally vanishes in no time at all.
Let the shoes air dry again after being cleaned. To restore the luster of the jazz shoes use black shoe polish (or tan for tan shoes). Take an old rag and dab a bit of polish on the rag. Rub the polish into the shoe by using small circular strokes on the leather. Work the polish into the entire leather shoe. Your shoes should shine when you are done.
Remove the laces of the shoes (if your jazz shoes have laces) and fill a sink halfway up with water. Let the laces soak for five minutes. Use hand soap and a small scrub brush to scrub the laces clean. Rinse with clean water and let hang dry. When dry, re-lace your shoes and you are set.