It could be something plugging the line somewhere. If you used the standard cheap install kit with a needle valve tying into your water line, then that could be the problem. Screw it in to pierce the hole in the pipe (maybe you didn't go far enough in to get a big enough hole) but also don't forget to screw it back out for the water to flow. Unhook the line from the refrig to see what kind of flow you get (that should tell you if the problem is in the water line side or in the refrig side). If that doesn't get it, install a tee in the water line with a ball valve for a shutoff. Then solder in the line to the refrig with a reducer (assuming it is copper). Think about a filter kit to help the taste. Good luck. Post back if that doesn't do it.

Also, it's very easy to cause a crimp in the water line when you push the refrig back onto place after the installation. Make sure the line's not crimped or crushed.

Don't forget that water dispenser is only using a 1/4" OD waterline, water is not going to come out of it like you kitchen sink.