Viability of an LED grid
It depends. Are the lights for accents or illumination? How densely you mount the LEDs and how many you use will also be a factor. The room size, mounting location, and intended use are all important as well.
Based on the experience I've had with LEDs, probably not. Only if you are using super bright LEDs should you need a heatsink. The datasheet should indicate if one is suggested or not. I know certain LED devices for photography that are made you be bright and left on for extended periods of time will use them. However, the typical LED strips and similar devices sold by many of the electronics and hobbyist retailers do not require a heatsink. The warmest component is likely to be the power supply.
I built a poker table with LEDs in it. My first instinct was to get a frosted glass or acrylic type of material to diffuse the light. I found it was still pretty easy to distinguish the individual LEDs. On a whim, I went to a local sign shop, and they suggested white polycarbonate. That is what I used, and it can be seen at the website I linked. Underneath I used an LED strip. I am still able to tell that the strip is in the middle of each partition, but the diffusion is much better in person than is shown in the picture. That said, a white, acrylic material (Plexiglass) would diffuse the light just as well, but the polycarbonate is stronger and has a better surface finish.


All in all, I would look at LED strips or pre-built strands of LEDs and space them apart evenly to fit the width of your panel. The power you need depends on the specific application, but it's better to get brighter lights than you need and dim them with a microcontroller and/or LED driver if you can afford it. Heat shouldn't be an issue unless the LEDs are tightly clustered and very bright. Check the datasheets. My project had no issue. Look around for a thin, white plastic material to diffuse the light evenly.

Edit:
If you do go with the LEDs linked in the question, Cree has an application note on thermal management.