Unfortunately, you can't use Marriott cheques at a Ritz Carlton. I have heard of unique cases where a Ritz will accept the cheques, but I've never encountered it so I'm not sure if it's really true. Every Ritz that I have asked has said that they don't. You should call the property you are traveling to in order to confirm with them. HOWEVER, you can use Ritz Carlton gift cards at any Ritz and those are available for purchase with your Marriott points - go to redeem points --> shop online --> search for Ritz in skymall. The value isn't nearly as good, but it's still better than spending your own money, right? Here is the comparison below to decide whether or not it's worth using your points now or saving them for Marriott money.
$1,000 Marriott Cheque - 135,000 points: Value per point = $0.7407
$1,000 Ritz Gift Card - 190,000 points: Value per point = $0.5263
Hopefull this helps!