So, I have a tendency to eat my organic produce without first giving it a thorough rinse. I mean what harm can a little dirt do, right? However, a friend mentioned that organic produce is affected by pesticide leakage from neighboring conventional farms and thus not truly organic or toxin free. Is there any truth behind this?

i don't know if there's any truth to that, but it seems like a likely scenario.

Still, even if a fruit or vegetable was grown without pesticides for sheer minimum health reasons you should wash your produce before eating for the same reason that you should wash your hands before eating.

Bacteria and viruses that are common in our environment (as in they are on every surface you regularly touch) can make you ill if your immune defenses are down. Stapholococcus, and Strep are just a couple of examples...and then there's E Coli, which you might find on any surface in a bathroom.

The rules for certified organic produce under the National Organic Program (NOP) require that organic fields "Have distinct, defined boundaries and buffer zones such as runoff diversions to prevent the unintended application of a prohibited substance to the crop or contact with a prohibited substance applied to adjoining land that is not under organic management." I do not know if these buffer zones are in fact sufficient to prevent pesticide leakage from other farms.
I agree with greeniac13 that you should wash produce anyway for the reasons he stated.
On another note, certain kinds of chemicals are allowed under organic regulations. For example, I worked on an organic farm one summer and they sprayed copper on some of the plants. In the scheme of things copper probably is not too harmful, and there are strict rules about how many days before harvesting the spray must take place, but this is another example where you probably want to wash your produce.

You can find all the regulations for certified organic products at GPO Access
Also check out the National Organic Program website.

Soil seepage on organic foods is only part of the concern in my opinion -- to me a huge issue is herbicides, pesticides, or other contaminants that have blown from elsewhere.

My understanding of organic is it must be grown as organic -- which doesn't actually guarantee that it is contaminant free.

A while back Consumer Reports compared organic produce with conventional produce. Organic had lower amounts of pesticides, etc., but it was not always pesticide free.

I wash all my veggies and fruits - just in case.

I agree that herbicides, pesticides, etc. blown from neighboring farms can be a problem. I know a farmer who argue over this with his neighbor, also a farmer, because the neighbor would spray when the wind was blowing the wrong direction and the chemicals would get into the organic farmer's fields. Communication is very important- between farmers and their neighbors to prevent these accidents and between farmers and customers so that every one knows what the vegetables have been exposed to.
It is somewhat analogous to second hand smoke, if other farms use pesticides and synthetic fertilizers then it will effect everyone, not just the people who eat that food. I guess then it becomes a policy issue. What chemicals should be allowed for use on any farm, organic or conventional? We tend to take for granted that our food will be grown with chemicals unless specially grown and marked (organic), but maybe it wouldn't have to be that way. Also it makes me wonder, if I cannot grow certified organic food on my farm because the neighboring farm is conventional and they spray a lot of chemicals, do I have any recourse under nuisance law (or something else)? Does anyone know how this works in practice?