The principle that God knows even the thoughts of our hearts is widespread in the scriptures. In modern days the Lord has said:
D&C 33:1
1 Behold, I say unto you, my servants Ezra and Northrop, open ye your ears and hearken to the voice of the Lord your God, whose word is quick and powerful, sharper than a two-edged sword, to the dividing asunder of the joints and marrow, soul and spirit; and is a discerner of the thoughts and intents of the heart.
The same conception is found in the Old Testament:
1 Chr. 28:9
9 And thou, Solomon my son, know thou the God of thy father, and serve him with a perfect heart and with a willing mind: for the LORD searcheth all hearts, and understandeth all the imaginations of the thoughts: if thou seek him, he will be found of thee; but if thou forsake him, he will cast thee off for ever.
(See also Psalms 94:11, John 2:24-25, Alma 12:3, Alma 12:14, Helaman 9:4, 3 Nephi 28:6, and D&C 6:16 for related concepts)
In Biblical language, the heart is the seat of thought (as opposed to the modern Western symbolism of emotion). It is in the heart that “true thoughts” are stored, thus knowing the “heart” and knowing the thoughts of men are the same thing.
Socio-religious: One of the things that this verse tells us is that vocal prayer was common. It is probable that this was not the personal vocal prayers that one might say, but a communal prayer, with multiple supplicants. Thus one of the things we learn is something of the more typical order of prayer for Alma’s people. It would appear that they would gather together for vocal, communal, prayer.
It is this communal prayer that Amulon forbids, though it clearly might extend to any individual vocal prayer. The banning of communal prayer has more behind it than simple vindictiveness. By disrupting the communal prayer, Amulon also disrupts other possible communal responses, such as organized resistance. By removing an occasion when the people were gathering together and bemoaning their fate, and asking deliverance, Amulon has removed a major brewing threat to his leadership - or so he would suppose.
As Mormon notes, the prohibition against public communal prayer did nothing to prohibit the hearts of the people, and the prayer of these righteous people went up to God even when they were silent.