[noun] + を顧みず、[sentence] [noun] + も顧みず、[sentence]
を顧みず goes after a noun and means "without regard for it": someone acts with no thought for a danger, a cost or other people's feelings. かえりみる顧みる is "take into consideration", and 顧みず is its old negative, "without considering". It is formal and written – news, speeches, biographies.
The noun is something a person would normally weigh: danger, health, one's own life, other people's objections or inconvenience. In conversation you would say をき気にせず or をかんがえ考えずに. Don't confuse 顧みる with かえりみる省みる, read the same, which means "reflect on your own conduct".
With も instead of を, it means "not even": the thing ignored is something anyone would have expected the person to think about.
The same words admire courage or criticise selfishness, and only the context decides. Risking yourself to save someone is praise; ignoring the neighbours is blame.