Yes, the rule exists and dictates that any vowel except for [у] that follows a soft consonant in an unstressed position is reduced to something between [э] and [и]:
Source: М.В. Пано́в. Ру́сская фоне́тика
In particular, this makes the words чистота́ (cleanliness) and частота́ (frequency) homophones while the respective adjectives чи́стый (clean) and ча́стый (frequent) have distinct pronunciations because the а in ча́стый is stressed.
In Russian [й], [ч] and [щ] are always soft. Other consonants are soft if followed by я, е, ё, и, ю, or ь. For the purposes of applying this rule we can ignore и (because it becomes a weak и but still an и), ю (because it gives the sound [у] which is an exception) and ь because its not a vowel. So practicality speaking we should be looking out for words containing unstressed я and е, including those at the beginning of a word.
The letter ё is almost always stressed. It can be unstressed in compound words e.g. трёхмерный (three-dimensional) which can be said without a secondary stress in which case the rule in question will apply.