There has been thorough scientific research done on this question. The main problem with Alton Brown's recommendation is that his room temperature "rest" is not long enough, since the scientific literature recommends 24-72 hours at room temperature, depending on acid concentration.
The most common acids used in mayonnaise recipes are acetic acid (vinegar) and citric acid (lemon juice). Vinegar is more effective at killing bacteria, but either can be used. Both acids are less effective at refrigerator temperatures than at room temperature.
For a general review of the literature, as well as specific advice on lemon juice (along with lab results), see this article. The take-home message, which you can see from the data (Table 1 in the link), is that the only successful sterilization of the mayonnaise that eliminated Salmonella bacteria occurred with at least 24 hours at room temperature with reasonable amounts of lemon juice.
The authors determined that the minimum quantity of lemon juice necessary to achieve this result was 20 milliliters (about 4 tsp) per egg yolk for 72 hours at room temperature, or 35 milliliters (about 7 tsp) per egg yolk for 48 hours.
Room temperature was necessary for sterilization: even after a week, all samples stored at refrigerated temperatures still tested positive for Salmonella, even with a high acid content. On the other hand, after a week, no samples at room temperature tested positive, even those that contained only 10 milliliters (2 tsp) of lemon juice.
A previous study concluded that vinegar was also effective (and more effective than lemon juice). Subsequent studies have indicated a dose of at least 20 milliliters (about 4 tsp) of standard white wine vinegar should be effective at 24 hours at room temperature -- though 72 hours was recommended if possible.
Other studies have shown that the addition of garlic and/or mustard will increase the sterilization effect, while salt inhibits it. The type of oil also matters: as documented here, "olive oil with garlic or basil showed the fastest rate of death, followed by soya, grapeseed, rapeseed, groundnut, sunflower, hazelnut and a blended olive oil."
It must be emphasized that all of these studies, without exception, recommend 24 hours at room temperature MINIMUM for effective killing of bacteria.
Before concluding, I suppose I have to address the mistaken assumption that mayonnaise is unsafe at room temperature in general. It is well-known among food safety experts that commercially prepared mayonnaise is perfectly safe at room temperature (for example, see the quotations at the end of this helpful brochure). The acidic environment and previous processing stages are plenty to keep mayonnaise safe -- you refrigerate mayo to keep it fresh longer, not because it is unsafe at room temperature.
In fact, adding sufficient quantities of mayonnaise to meat-based dishes like chicken salad or ham salad can actually slow growth and even kill Salmonella and E. coli on the meat and thus make the dishes safer. In the linked study, meat salads with mayonnaise held at room temperature for five hours had very little growth of Salmonella, compared to what would be expected without mayonnaise. Note that refrigeration in such cases is still recommended, since the meat pieces can still spoil at room temp and the acidic sterilizing effect of mayonnaise is somewhat diluted when mixed with other things. (Mayonnaise--along with any liquid or semisolid food--when handled improperly around contaminated food, can lead to cross-contamination of bacteria in mixtures that dilute its sterilizing effect. But that's not a property unique to mayonnaise at all.)
In the past, homemade mayonnaise did not have the necessary processing to render it safe, which is the impetus behind the many studies I've cited here. These studies show what you need to do to make it safe. After 24-72 hours (depending on factors listed above), the homemade mayonnaise can be safely refrigerated to maintain its quality for longer than storage at room temperature.
[EDIT: One additional corollary to the research above is that homemade mayonnaise is actually the most dangerous when it is fresh. I've occasionally heard people say, "I make homemade mayo, but I always use it right away, so it's safer." In reality, as discussed in the linked articles, the acid will stop Salmonella from further growth and eventually cause it to die off in undiluted mayonnaise, even with significantly less acid than the recommended quantities. Most egg-based food poisoning is caused by contamination from the shell, where Salmonella bacteria is commonly found and will grow once it comes in contact with a liquid medium. This requires time, so fresh egg dishes are generally safer. (Note that in the U.S., unlike almost everywhere else in the world, eggs are washed and their exteriors are disinfected, so Salmonella infection from egg shells is much rarer.) With mayonnaise, though, the growth of small numbers of shell bacteria accidentally introduced into the mixture is inhibited by the acidic conditions. Instead, the concern is the much smaller number of eggs (estimates usually say about 1 in 20,000) where Salmonella is present in the interior of the egg, and the liquid medium may already contain a high enough population of bacteria to make someone sick. Those rare types of eggs will make you sick even when eaten fresh, which is the reason some restaurants have warnings about sunny-side eggs or runny omelets. The average person will only encounter one of these internally infected eggs a few times in his/her lifetime, so the risk is pretty small. But considering that thousands of people probably make homemade mayonnaise worldwide every day, these rare eggs will still occasionally cause sickness. While the acidic recommendations above are well-researched, I personally would still use pasteurized eggs in making homemade mayonnaise for children, elderly people, pregnant women, big parties, etc. On the other hand, if you eat (cooked) runny egg yolks on a regular basis, you shouldn't be concerned about mayonnaise, which actually has a lower risk if it sits for any period of time.]