Yes, Chipotle does take American Express credit cards, along with Visa, Mastercard, and Discover cards. Chipotle Mexican Grill also accepts prepaid cards, debit cards, cash, and Chipotle gift cards as valid payment methods. Finally, Chipotle takes digital wallets such as Apple Pay through the Chipotle app.
You can't pay for Chipotle using a check or money order, and it's unlikely that you'll be able to use digital wallets in the restaurant. Regardless, you shouldn't have any trouble paying with an Amex credit card at Chipotle.