I have a 99 civic ex and when I start the car (any time of day), put it into drive and press the gas pedal it grinds once, and only once.
I have no idea what it could be. Ive had two different transmissions in there and the noise is still there.
I don't hear the noise when I'm at a dead stop and accelerate, only when I have the car off, start the car, then accelerate.

 

This is purely a guess but the one thing that is different from car off, start and go and car running then going from stop is the Starter.

I would think a sticky soleniod on the starter would make a lot more than just a one time grind noise but it is the one thing that is different between the two situations you mentioned. If anything, I would think it would be near constant once the car fired up.

If you've had different transmissions with no change then it's either motor, clutch or axle issues I would suspect.

 

That's definitely a good start. I'll check my started and go from there. I do have an extra one I can put on to try.
Thanks

 

Changed the starter, and still have the grinding noise. I don't even think this starters any better than the one I had lol