Many books record a custom to recite "Lamenase’ah Binginot Mizmor Shir," the 67th chapter of Tehillim, during Birkat Kohanim throughout the period of Sefirat Ha’omer. The custom is to recite this Psalm according to the structure of a Menorah (as depicted in many Siddurim). It is recorded that observing this custom protects a person from harm throughout the year and brings him success in all his endeavors.
This custom has given rise to a discussion among the Poskim (Halachic authorities) regarding the propriety of reciting Biblical verses during Birkat Kohanim. The Shulhan Aruch indicates that one should remain perfectly silent during Birkat Kohanim, except for the quiet response of "Baruch Hu U’baruch Shemo" and the audible response of "Amen." Seemingly, then, it would be improper to recite a chapter of Tehillim during the recitation of Birkat Kohanim. Nevertheless, several Aharonim (recent authorities), including the Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909), record the practice to recite this chapter during Birkat Kohanim.
However, Rav Daniel Frisch ZT"L, in his work Usfartem Lachem (listen to audio recording for precise citation), notes that this practice deviates from the authentic custom as recorded in the earlier sources. The custom originates from the writings of the Roke’ah (Rabbi Elazar of Worms, Germany, late 12th-early 13th century) and other sources, which mention reciting this chapter of Tehillim after the recitation of Birkat Kohanim, and not during Birkat Kohanim. Indeed, this is the version recorded in the writings of several Aharonim, including the Shalmeh Sibur, Mo’ed Le’kol Hai (by Rav Haim Palachi), Shebet Musar and others. However, as Rav Frisch points out, some Aharonim did not have access to the Roke’ah and other primary sources, and recorded the custom based on what they saw in secondary sources. Their account of the custom was thus imprecise, as it mentioned reciting "Lamenase’ah" during Birkat Kohanim, rather than afterward. But now that we have access to the original sources, Rav Frisch writes, we should follow the authentic practice of reciting this chapter of Tehillim during the Sefirat Ha’omer period after Birkat Kohanim, as the Hazan recites "Sim Shalom," and not during Birkat Kohanim.
Summary: There is a custom to recite "Lamenase’ah Binginot Mizmor Shir" (Tehillim 67) after Birkat Kohanim throughout the period of Sefirat Ha’omer, and this practice helps protect a person from harm and guarantee his success. Some sources record a custom to recite this chapter of Tehillim during Birkat Kohanim, but the correct practice is to recite it after Birkat Kohanim.