96 neon/electrical problem

I have a 96 neon. The battery in the car kept dying and I had problems starting the car in the morning. I took it in to my mechanic, had the alternator checked out, it's fine, and the battery is good (about a year old). He was thinking the remote starter was draining the battery, so he removed it. Now, this morning, the same thing happened. Does anyone have any ideas on what else could be draining it? Where should I start looking?

Thanks!
Andrzej