Is it possible to skip this step, and use milk straight from the fridge? Would the result be terribly different?
The biggest reason to heat milk to almost boiling before fermenting is that it improves the texture of the yogurt.
During fermentation the bacteria consume lactose and produce lactic acid which causes the milk proteins to denature and coagulate trapping most of the fat. The proteins involved are primarily the casein proteins.
When this happens, there is still quite a bit of protein left that isn't bound up in the new casein mesh. All of the albumin proteins are water soluble and will not add to the structure of the yogurt.
These albumin proteins denature when they are heated. For this reason recipes universally call for the milk to be heated to 190 and then cooled. The albumin is denatured and is able to tangle up with the casein during fermentation and add to the yogurt structure.
Skipping this step will make a very profound difference to the structure of your yogurt. Without it your yogurt will be thinner and much more fragile. When you scoop it there will be more whey and all that albumin will wash out in it.
According to Harold McGee (On Food and Cooking pp 48)
"These treatments improve the consistency of the yogurt by denaturing the whey protein lactoglobulin, whose otherwise unreactive molecules then participate by clustering on the surfaces of the casein particles. With the helpful interference of the lactoglobulins, the casein particles can only bond to each other at a few spots, and so gather not in clusters but in a fine matrix of chains that is much better at retaining liquid in its small interstices."
Casein is the milk protein that gels to form yogurt, encapsulating whey in a "spongy" matrix. Casein floats around in milk in the form of globules, or micelles. In fresh milk, the suspended micelles bump into each other and bounce away, going off in different directions.
When an acid is added to the milk, the interactions between the protein micelles are modified, and now instead of bouncing off each other, they stick together when they meet.
Now here is the reason why you need to heat the milk and failing to do so results in sub-par yogurt:
As the milk is heated, the microstructure of the protein micelles changes, they become bumpy. Micelles in unheated milk do not have these bumps.
The bumps on the micelles make it so only a limited number of sites on its surface are available to interact with other micelles. This means that the micelles can only adhere to each other in branched chains.
In unheated milk, there are no bumps to prevent adhesion, so ALL sites are available, and instead of forming a spongy matrix, the proteins form a curd. This means that less liquid can be trapped in the "pockets" of the matrix. Therefore, you will have runnier and grainier yogurt.
Check out this link for technical details and diagrams: http://www.medicinalfoodnews.com/vol01/issue5/kalab.htm
There's a specific culture of yogurt that can be fermented at room temperature without keeping the milk warm that produces the desired texture. The "Caspian Sea" yogurt strain works well with room temperature fermentation and, in my experience at least, required no heating.
I was able to obtain this strain a few years back from friends who were growing it; it works as long as you're regularly using and replenishing it, but it can spoil eventually. I believe you can find vendors for this kind of product at health food stores or online, but I don't know how you'd be able to distinguish quality or efficacy, so I can't speak to the best way to obtain the culture.
Keep in mind, though, that this is a much more viscous and lower-acid yogurt than you may be used to. Because of these characteristics, it became quite popular in Japan and among Japanese immigrants to the US, which is how I first encountered it.