The J train is a great way to get downtown from JFK too - no need for a cab.
The subway in general is good to get to and from JFK. I've used it on all my trips and found it to be the best value option, costing a standard fare + $5 for the air train (compare that to LHR!). But I've never had too much luggage. Going into the city is never a problem, there's always been plenty of room on the A and E trains (never tried the J/Z option). Coming back can be more difficult because you'll probably be boarding in Midtown, the busiest part of the network - you will not be popular if you're taking up multiple seats with big suitcases during the rush hour!
I've been to NYC twice and both times used the subway as almost my sole means of transportation, including airport transfers. I had no problems at all during either of my visits. It is a fantastic system.