I. Walking with God
Are you allowed to text or check your e-mail while praying or reciting a blessing? The Taz (Orach Chaim 191:1) writes in strong language that you must pay full and complete attention while reciting a blessing. He compares it to "walking contrary with [God]" which can also mean walking with God carelessly. This leads to a punishment of "And I will walk contrary with you" (Lev. 26:23-24). This seems to conclusively answer the question but it is actually more complicated.
II. The Shema Exceptions
The Mishnah (Berakhos 16a) states that workers up in a tree can recite Shema while still in the tree but must descend to the ground to pray the Amidah. Rashi explains that the Amidah requires extra concentration, which is not possible while up in the tree. The Gemara continues that workers must stop their work for the first section of Shema (commentators disagree exactly what constitutes this section) but may continue their work for the second section.
The Gemara (Yoma 19a) tells of a time when a talmudic sage was quoting a Mishnah and mispronounced the name of a rabbi. Rava, who was reciting Shema at the time, motioned with his hands to indicate that the name should be pronouned with a "beis". Shouldn't Rava have been concentrating on the Shema and not motioning to others? Not to worry, explains the Gemara, because he was reciting the second section of Shema. If the Shema does not require great concentration and you can work and motion to others while finishing it, perhaps you can also text and check your e-mail.
III. Two Reasons for Focus
The Ramban (Milchamos to Berakhos 19a) distinguishes between two reasons that would forbid motioning or working while reciting Shema: 1) the need to concentrate, 2) the obligation to have kevi'us (fixedness). The former is a function of intent while the latter is a matter of respect and context. The second section of Shema requires neither.
I suggest that this also applies to the blessings before and after Shema. The Maharsha (Yoma 19a) connects the issues of kevi'us and greeting someone during Shema. Since the permission to greet or respond to a greeting at specific points in the Shema apply also to the blessings, the lack of needed kevi'us presumably also applies. The language of the Rambam (Mishneh Torah, Hilkhos Keri'as Shema 2:4) also implies that the blessings of Shema have the same status as the second section
What seems to emerge, then, is that while you may not text or check your e-mail when you are reciting a blessing or the Amidah or the first section of Shema, you may do so while reciting the blessings and the second section of Shema.
However, perhaps we have to distinguish between performing repetitive work with your hands or motioning with your hands and doing something that requires thought, like writing or reading. The Ritva (Berakhos 19a sv. lo) writes: "And even though you do not need actual intent, you still cannot be involved in other things so that you have a rested heart." The Shulchan Arukh Ha-Rav (183:14) only allows mindless work that you do with your hands and nothing that requires concentration. Therefore, it seems that you must avoid your electronic communication devices while praying, reciting blessings and saying Shema.
As always, ask you rabbi about halakhic matters and don't rely on what you read online.
UPDATE: R. Ezra Bick wrote this comment:
It is prohibited to read email or text during any bracha or any part of shema.
The Gemara is concerned with someone working, who has an obligation to his employer, and therefore descending from the tree is problematic. The same applies to a short response in order to correct a mistake, or even to greet someone, which, under certain conditions is a mitzva. However, doing one's own business while davening is not even worthy of a question. Despite the halakhic disclaimer at the end of this post, I think it should be corrected. This is a very high quality blog, which might lead someone to take this post seriously, meaning l'maase, which would be a travesty of halakhic living.