Would diesel (Jet A) cause less pollution than other fuels?
This is a good question which does not have an entirely straightforward answer as the different types of pollution produced and their effects on the environment are still being studied. Also, I know of no actual studies of aero diesel versus avgas versus unleaded avgas pollution in different types of aero engines to base a conclusion off of. I will draw some parallels with auto engines and fuels as they are similar in many ways. 

Comparing the efficiency and pollution of a modern aero-diesel engine to the archaic avgas engine technology in most light airplanes is not really a fair comparison because modern aero-diesel engines have computerized electronic fuel injection, and often computerized variable pitch propellers which combine to create the best efficiency for the phase of flight. Old engine technology has no electronically controlled fuel injection (often no fuel injection at all), and often they don't have the gauges installed to inform the pilot how best to lean the engine. A better comparison would be a modern aero-diesel to a modern avgas engine with similar levels of technology, that's something I'd like to see. 

There are many different types of pollution, and the different types of engines produce differing amounts. I will list the major types: 


CO2: If you look at it from a CO2 perspective then auto diesels are less polluting than auto gasoline engines, which is why in many places across the world people have been moving to diesel. Governments (many in Europe) have in some cases taxes diesel less to make it more attractive. Whether aero-diesel engines produce less CO2 than a comparable technology avgas engines is not certain, as I said I've seen no actual data. Aero-diesel engines are heavier than avgas engines because they are partly made of steel, but they don't need magnetos which saves some weight, but then they are turbocharged which takes weight. Looking at the auto comparison I'd say aero diesels are probably better on CO2
NOx compounds: Nitrogen oxides are linked to asthma and breathing disorders, and diesel engines produce more of these than gasoline engines
Sulphur compounds: These are also linked to asthma and breathing disorders. Jet fuel has higher sulphur content than diesel fuel, and much higher than gasoline/avgas. Sulphur compounds may actually be helping to cool the planet though, as they help form clouds (acidic clouds) which reflect sunlight back into space
Fine Particulate Matter (FPM): particulates are a biggie from an environmental and health perspective these days. Particulates are linked to asthma and breathing disorders, and there is some evidence that black carbon particulates are heating the planet and causing glacier melt at an accelerated rate as they absorb heat. Diesels produce much more particulates than gasoline engines do


There is growing scientific evidence that the health and environmental costs of particulates, nitrogens and sulphur compounds of diesel outweigh the benefits of lower CO2 levels. I know in the UK the government is thinking about a U-turn on their stance on diesels, and many people are looking at going back to gasoline engines. 

The biggest polluter with avgas is the lead though, in the form of Tetra Ethyl Lead (TEL) which is extremely toxic and very expensive to deal with. Unfortunately, without the lead avgas engines can rip themselves apart because of detonation at high density altitudes. This is an edge case to be honest, most light aircraft will never be in that situation, but it is potentially deadly in enough cases that you can't just get the lead out (pun intended). The lead makes avgas much more expensive than Jet fuel, which is one of the drivers behind aero-diesel; it's cost, not environmental benefits which make it attractive.