We do not say Tachanun all month.
After saying the Nassi (p. 368), we say the Yehi Ratzon (p. 371) which is printed in Torah Or Siddur. Kohanin and Levi'im also say it. [1]
Tuesday Nissan 1, Rosh Chodesh 5703
Chumash: Tazria, Shlishi with Rashi.
Tehillim: 1-9.
Tanya: But as for him (p. 193)...the Esoteric Discipline. (p. 193).
My father instructed his brother-in-law, R. Moshe Horenstein, a Kohein, to say Yehi Ratzon after the Nassi, noting that even a Kohein or Leivi must say it, for it is related to Ibur. [2]
Notes:
The Nassi describes the offering brought by the Prince of each tribe during the twelve days of dedication of the Sanctuary.
See Glossary to Tanya I. Ibur is "gestation," the undifferentiated state prior to the appearance of organs. In the Yehi Ratzon prayer, we say, "..if I am of the tribe of .....," indicating the possibility of any of the twelve tribes (including Efrayim and Menasheh; excluding Levi). This would seem inappropriate to a Kohein or a Leivi, who are by definition from the Levi tribe. However, the concept of Ibur indicates an initial state of total merger, coalescence - in this context the coalescence of all souls. The Kohein then, is not absolutely unrelated to the other tribes of Israel.