you shouldnt have any problems checking through to the final destination. although you will still need to collect boarding passes.

one word of caution - if you are travelling from Asia to the US and your connecting AA flight happens to leave earlier than the departure time of your CX flight make sure they put the right date for the connecting flight.

and as rightly mentioned above, you will need to claim your bags at the first point of entry into the US and clear customs. but at most major airports AA will have a baggage re-check area right after customs - and your bag will already have the tags attached so you jst put it onto the belt.