One problem related to power train has been reported for the 2003 Mercury Grand Marquis. The most recently reported issues are listed below. Please also check out the statistics and reliability analysis of the 2003 Mercury Grand Marquis based on all problems reported for the 2003 Grand Marquis.
Rear axle and bearing noises. Marquis bought new. Owner's son-in-law noted minor bearing noise in rear when visiting in 2004. Son-in-law received car after death of mother-in-law in 2012, car had 24k miles at that time. Noticed pronounced rear end noise at that time (a whaa-whaa-whaa matching speed changes). Left rear axle and outboard bearing found to be pitted and were replaced by independent repair shop. Research by owner indicates that axles for this, and other similar Mercury and fords had factory axles installed that had not been properly heat treated. Now, Nov. 2014, at 43. 4k miles the right axle is starting to exhibit noise similar, so repair is up-coming. Will examine NHTSA recall 04v328000 as noted here in other comments posted. Also reference Ford/Mercury service bulletin 0355.