The Shulhan Aruch (Orah Haim 183:9) rules that one must sit down with a sense of reverence to recite Birkat Hamazon, even if one was previously standing or walking. Furthermore, it says (ibid:12) that one is forbidden from performing any acts of labor while reciting Birkat Hamazon. The Mishna Berura (O.H. 183:37), citing another Halacha in the Shulhan Aruch (Orah Haim 191:3), clarifies that one must stop what one is doing while reciting not only Birkat Hamazon, but rather for all blessings.
The Ben Ish Hai (Year A, Parashat Hukat, § 8) says that one should not only refrain from outright acts of labor, such as cutting wood and the like, but even seemingly benign acts, as it shows a lack of reverence for the blessing being recited. Examples of such acts given by the Ben Ish Hai include drying one’s hands, putting on clothing, waving or hinting with one’s eyes or lips. The Taz (O.H. 191) says that one should not even engage in Torah study, such as contemplating a Torah thought or looking in a book, while reciting a blessing. Interestingly, Rabbi Yosef Haim Sonnenfeld (Shu”t Salmat Haim, § 155) permits folding of one’s Talit during the Shir Shel Yom or later parts of the prayer.
Parenthetically, in light of the aforementioned Halacha, one may question the Shulhan Aruch’s (Orah Haim 63:7) ruling that one must stop one’s work to recite the first paragraph of Keriat Shema, but may continue working while reciting the second and third paragraphs. The Pri Megadim (Mishbetzot Zahav, § 63:1) explains that the essence of the Mitzvah of Keriat Shema is contained in the first paragraph, while the last two paragraphs have more of a status of Torah study. Performing labor is permitted while learning Torah and this explains the leniency regarding workers who may continue to work while reciting the second and third paragraphs of the Shema.
Summary: One should not be involved in any other acts while reciting a blessing.