According to mineralogist Hershel Friedman, dolomite has a Mohs hardness of 3.5 to 4. The Mohs scale is a scale of hardness measurement for rocks. Talc, the softest of all rocks, has a hardness of 1. Baby powder is made from talc. Diamonds, the hardness mineral on earth, has a Mohs hardness of 10. Since dolomite is 3.5, it's a lot closer to talc. For this reason, dolomite is an easy rock to cut and shape.
Before you cut the rock, observe the striations and multicolored streaking patterns. Think about the final piece of jewelry you are making and how the pattern would look if cut parallel to the grains or perpendicular to the grains. Decide on the cuts, and draw saw lines on the rock with a marker. A circular saw with a diamond blade can only cut straight lines.
Holding a small stone with your fingers near a spinning saw blade is never a good idea. Special holding jigs which have a claw-like tip hold the stone firmly. These jigs have a handle so your fingers are not near the blade. Make a push stick bout 12 inches long out of a 3/4-inch dowel. Use the push stick to push the rock against the saw blade. Since dolomite is a soft fragile stone, light pressure is all that's needed.
Thousands of different types of tile cuttings saws exist in the world. You do not need a huge industrial saw. A small saw that looks like a table saw will suffice. The only glitch is that tile saw blades, because they are thick, tend to leave a very wide kerf, which is the width of the gap left by the saw blade after cutting. A kerf, in essence, is wasted material; the less waste, the better. Lapidary saw blades are very thin and do not remove a lot of material, resulting in a narrow kerf. Lapidary saw blades are plentiful and readily available from jewelry and gem cutting supply houses. You can buy a special gem cutting saw, but these tend to be too small for large lapidary work and more expensive than tile saws.