We are having a three season room installed. It will be one to four feet above ground(slope). We are replacing a deck. The deck had plastic covered with gravel under it.

1. Do I need that under my three season room?

2. Will solid sheeting around three sides of the bottom cause a moisture problem?

Your three season room should have the following.
1. Yes I do suggest that you put new plastic under with a gravel top to help protect the plastic.
2. Solid sheeting is used all the time as a skirt around decks and rooms such as this. However you should install screened vents to allow for air to travel under the deck or your concern will come true moisture will develop.
3. I also suggest that you insulate between the floor joists before you place the floor down.
Protect this material under the deck area with 1/4 inch plywood to prevent nesting of mice and other critters who love to live in this.
The insulation will make the floor more livable during the cooler months you use the room.

Also as I have said on some other posts in the past. Many folks end up using this type of room year round. Doing a few upgrades now such as insulation in the hard to reach spaces can lower the conversion cost should you decide to heat this space for year round use.