As mentioned elsewhere on this site, we don't say tachanun during the entire month of Nissan, because a majority of the month is spent "in kedusha," skipping tachanun -- therefore we omit tachanun for the entire month.
Don't we do the same during Tishrei? Based on the above reasoning, why should we say tachanun at all during the month of Tishrei? (assuming that we don't say tachanun after Sukkos)
Nissan starts without Tachnun because it is the time when the Mishkan was inaugurated - that is the first 12 days. Then the 13th is the Isru Chag of those 12 days (The Tzemach Tzedek quoting the Maharil), and then there is no Tachnun on the 14th because the Korban Pesach was brought, and then Pesach. So there is no opportunity to say Tachanun until the majority of the month is already without Tachnun. (See Shulchan Aruch HaRav O.C. 429:9).
In Tishrei, that situation only exists when the end of Succos is reached, so it would seem that the rule that once you don't say Tachnun for the majority of the month you don't say it for the rest of the month only applies once the majority of the month has actually passed without Tachnun.
The Rav of our shul spoke about this when discussing our shul minhag of not saying tachanun until Rosh Chodesh Cheshvan. Note that this is our shul minhag as other shuls in the community start tachanun after Isru Chag (or maybe on Isru Chag, I am not sure).
The first part of the month continues tachanun because of the inyan of teshuvah and din. After Succos, we continue not saying tachanun because of the inyan of simcha and adding to Shmini Atzeres because it is a "Regel bifnei Atzmo" rather than the end of the chag (like the last day of Pesach). We also do not say tachanun for the week after Shavuos since there was a week afterwards that the korbon chagigah could be brought. Even though a separate chagigah was not brought on Shmini Atzeres (and the Ymei miluim of Succos are over) we still extend the simcha of the Chag.
This is from memory as to the explanation that our Rav gave for our shul minhag (which goes back to the founding of our shul as a separate minyon. Note that other shuls do not have this minhag (as I said earlier).
I do not recall the sources that our rav brought. Since there are those that do say tachanun, there is what to rely on in both ways.
