In the word giorno the letter i has essentially the same role as the h in che. We don't “pronounce” the i, which just denotes using the palatal sound for g like in gesto: /ʤ/. You can call it “silent”, if you wish, but its role is just “orthographic“: it's like a diacritic, but it's written next to the letter it modifies. Anyway, a silent letter seems to be one that stands for no sound, in Italian only h.
Depending on the speaker, the realization of the phoneme /ʤ/ can vary, but there is definitely no diphthong involved.
There is also no diphthong in già and giù; they are written with an accent for orthographic uniformity with più and piè, where a diphthong exists. A similar case is ciò.
This “orthographic i” is used after c or g to denote the palatal sound when followed by a, o or u:
ciabatta, rancio, ciuffo, giacca, giorno, giusto
Conversely, the “orthographic h” is used to denote the “hard“ sound when c or g are followed by e or i:
chela, china, alghe, fanghi
The i in cieco has a different nature: it's not pronounced in most of Italy, but it is (at least weakly) where the phenomenon of mobile diphthong is still alive (for instance the region of Naples). The “rules” of the mobile diphthong requires it appears in cieco but not in cecità. Notwithstanding the fact that most Italians don't pronounce the diphthong in cieco, the orthography maintains it.
The i is also “orthographic” when the trigraph gli is followed by another vowel: daglielo.