1989 Honda Accord Car cranks but won't start

OK, I'll try to give as much info as I can because this is confusing the hell out of me.

When I turn the key to position two, the dashboard lights come on but the fuel pump doesn't sound like it's coming on. There's plenty of cranking power when I turn the key, but the engine doesn't sound like it's getting any fuel.

I don't know if this is relevant, but:

This problem seemed to develop slowly during the last two weeks or so. The first start up on any given day (or after several hours of sitting in a parking garage) would be fine, but if I tried to start it in the middle of the day after driving for a while, I would get a spotty ignition--sometimes needing to pump the gas pedal to get the car to idle.

Usually, I would start the car first thing in the morning or the late evening, these "troubled starts" were usually during the middle of the day in 110 degree heat (I live in Arizona).

The ignition switch is fine--it was replaced by a mechanic a few months ago.

I replaced the fuel filter and things seemed ok for a day or so--then it wouldn't start again.

The fuel filter was replaced inside of two years ago, and while I think it *could* be the problem, I'm not sure how to confirm that.

The Haynes manual also suggests that the F/I "main relay" might be the problem.

Any advice?