I would guess try replacing fuel pump, But before you do , spend 20.00 bucks on a Haynes manual at your local Auto Zone or Advanced Auto and do a little multi-meter work before buying the wrong part. Everything is electronic so you never know. I had an older Ford Tempo once and the fuel pump did the same thing then eventually went out and had to replace. Hope this helps!!
Added by CamaroLuver: Kudos on the mutimeter advice! Better than going out and buying a bunch of parts, and doing the guess work. It COULD be the fuel pump then again could not. Just because you replaced the relay doesn't mean it's working right. Touch the fuel pump relay when you don't hear your pump come on, have a friend turn your car on (not crank) and see if you feel the relay click over. If not then you aren't getting power to your relay. Then it's a different problem entirely. Wiring, a ground, possibly computer problem, do a lot of testing before you go out and buy parts. Also test if you have spark before you rule out to be a fuel delivery problem (which I'm guessing it is) I learned that the hard way. Had the exact same symptoms, bought a new relay, went through 3 pumps before we realized it wasn't either one, got a new ignition coil and switch, ECM, fuel injectors. Turned out to be some bad wiring and a fusible link. Electrical problems are the greatest!!! :-p *sarcasm*