I have tile over cement and the grout is damp, why?

Is this a new installatiion of tile? If it is new then it is supposed to be damp while curing.

Concrete floors must be completely dry before applying thinset. However, they rarely are completely dry, and you may have damp grout forever. Did you test your concrete for 48 hours with a square of plastic taped down with duct tape? If moisture condenses under it (from below ground), then you can't count on tile lasting. Eventually it will all come off.

Tile installers swear by Ditra underlayment. I know nothing about it, and it's expensive, but supposedly worth it.

When did you grout? Over new or old concrete? If this is new concrete, did you use thinset or Acrylpro to set the tile??

Yeah, if this is an old tile/grout installation, you could be describing and issue whereby the WEEP holes in the shower drain/pan have become filled up with gunk and now the water in the pan isn't draining out and the moisture is showing up in the concrete under the tile and into the grout joints. There is a couple things you or a contractor may be able to do to resolve this, but there are no promises it will work 100%.

Let us know if this is old work or new work...;)

Mark