Hi there.
I got a 24 inch Full HD TV to use also as a monitor yesterday (it's a Sharp LC-24LE220E). It has a VGA port for PC and 3 HDMI ports.
I have as a graphics card an ASUS AMD Radeon HD6850 which has 2 DVI ports and 1 HDMI.

If I use a HDMI cable to connect to the TV I get a grainy picture.
If I use a VGA cable with DVI-to-VGA adapter I get a crystal clear picture.

Both at 1920x1080 @60Hz. On Win 7 x64.

Using the same cable and ports, when I connect my PS3, it looks ok as far I can see.
Anybody got any idea what the problem might be? Haven't been lucky with Google yet, I'm still looking.

if you use HDMI, the TV is more picky... have you set the resolution as the same as the TV is? This must be done. I don't know why, but the scaling using HDMI is much worse than the scaling with a VGA one.
oh, and just found out, that some TVs are trying to automatically improve the picture... this might be the case for you.

I don't know the Sharp, but LG has a setting for this to tell the Monitor that the input comes from a PC and then the TV turns off the auto picture improvement. After that you have a crystal clear picture.

I think something like that might be the problem but I don't know if there's a way to turn it off.
I mean if I select as input the PC-RGB port when I get into the Picture menu I have

Aspect Ratio with:
-Full
-Dot by Dot
-Normal 4:3

And also I have a "Screen" option in the menu with Vertical, Horizontal, Size, Phase options.

While when I select HDMI input I get in the Picture Menu:

Aspect Ratio with:
-Auto
-Full
-Underscan

And the "Screen" menu is disabled. Now I don't know if any of these options affects the quality but obviously the TV has different options for PC than other devices and I can't find any advanced options to turn something like this on or off. This is annoying. If I can't find another solution I guess I need to keep it on VGA and buy an audio cable to connect also the audio. Annoying...

Check the manual for any picture improvement settings and then turn all of them off. Usually the "improvement" settings do just the opposite. Especially if there are any scaling or denoising settings. I took a quick look and page 20 of the manual shows "Noise reduction", turn that off.

Yeah, that was one of the first things I tried. Noise Reduction has 5 options: Auto, Off, Low, Middle, High. I get the same problem no matter which I choose.
Also tried the Sharpness from 0 to 100. No use :(

I think I'll also try contacting support. See what they have to say about it.