I've been reading the responses to Joe's post and find them very interesting. I'd like to offer an observation from "the other side."
I've been research my family on and off for over 30 years (yikes!). There are times I am deeply into the hunt and spend hours and hours on nothing except genealogy. Then I have down times, where I lurk on message boards but not much else.
In my own research, I have very specific goals: I am interested mainly in direct line ancestry and branches that do not go too deep. So, I while I have information on my grandparent's siblings I have even less on their descendants. That is a choice I made long ago and feel that I have plenty of work to do to keep me busy. Given those parameters, if a 3rd or 4th cousin from one of these distant branches were to contact me, I would respond with whatever I might or might not know about how we are related, but I am not apt to engage beyond the initial back and forth emails to understand our potential connection. I understand that searching a learning about far distant relatives, while interesting and can, I know, lead to discoveries about "closer" relations, that is not my primary interest.
Another example. I have a distant connection to a gentleman who is related to some of my second cousins not by our shared ancestor, but though theirs. In speaking with this individual several months ago, I was kind of interested in trying to prove our relationship (he actually has two connections to my tree: one possibly to my 2nd great grandfather, the other is through marriage to a branch from a different great grandfather -- that is the branch I am not researching). So, I setup a "research tree" in Ancestry with just enough names, etc. to start searching for hints and documents to prove this connection and I shared this tree with this "cousin." While he did give me good information, he also keeps telling me about relatives that are beyond the scope of my research (spouses, their parents and their children and children's spouses, etc.). I listen to/read his feedback and thank him for the information, and leave it at that.
I am offering this to this thread only to shed some light on why you may not hear back from your supposed 3rd cousin twice removed. They may be wonderful people who are dedicated to their research. However, their research goals and interests may not (currently) intersect with yours.
So, please write to everyone you want to connect with. Take the wonderful advise given here about how to craft your emails. Just be aware that if you are casting a wide net, you may reach folks who do not share your research interests even if you do share some distant ancestor.
Karen Weinstock Gwynn
WEINSTOCK, PASS/PESIS, NUSBAUM
LURYE, GILBERT, BALANOV/BALANOVA/BALANOFF/BAILIN, LANSKY