Sand - small granular particles of insoluble stone. It is a mechanical means to keep your grip as it is hard, and cannot be compressed under your weight. It pits the less hard surface of the ice giving traction. It is useful under all temperature extremes. Avoid fine beach sand in favour of large grain sharp sand.
Salt - small granular particles of soluble calcium chloride or potassium chloride. It is a partly mechanical means and chemical means to keep your grip as it remains hard until it dissolves. After it dissolves, it melts the ice so that it can flow away to leave the surface wet but not icy. It is useful until the temperature falls below —3°C (28°F) when even salt water begins to freeze. The concentration of the salt is a factor. The more concentrated the salt the lower the temperature it is effective until you reach —15°—15°C where it becomes ineffective no matter the concentration. It's too cold to dissolve the ice and it sits on top of the ice layer. Salt is corrosive and can cause some staining which will wash away with fresh/rain water. It will become problematic with adjacent lawn or gardens.
Wood chips - small salt-treated chips of wood from saw cuttings or milling. Coarse flakes provide some mechanical means of traction. As it is crushed in use, it becomes slightly less efficient over time. It is good for traction down to —30°C (—22°F) The biggest benefit is that it is organic and will degrade naturally into mulch. Magnesium chloride is coated on the chips to increase the effectiveness of the stuff. It is more costly than plain salt but it covers a greater area for a longer time in colder temperatures.
Other things to consider are good old Kitty Litter (like sand), Ashes which are also biodegradable, and neither is so corrosive as salt or treated wood chips.
None of the above I would consider "dirty" but I think that all would qualify as messy to clean up.
I hope this helps with your decision which to use.