There’s no mnemonic – it’s a question of whether the French verb needs à. Envoyer and dire do – envoyer quelque chose à qqun and dire quelque chose à qqun, so your first two are wrong. The correct grammar is

Je leur ai envoyé…

Je leur ai dit…

But the causative doesn’t take à unless there’s also a direct object. So je leur ai fait comprendre la phrase but je les ai fait comprendre.

First, let me correct the phrases you mentioned:

• Je leur ai envoyé mon message
• Je leur ai demandé de ne pas me perturber
• Je leur avais fait comprendre la phrase

Now, let’s come back to your question: you are confused with leur and les.

There is one simple rule that makes a huge difference between them and which will help you to distinguish definitely between them:

1. Remember that leur is the plural of lui. It always has the role of complément d’objet indirect.
2. Les plays always the role of complément d’objet direct. Examples:
   • Je les ai mangés: I have eaten them
   • Elle les a oubliées: She has forgotten them

Rule of thumb (the other answers are more complete and will also explain the why):

Je leur ai fait une faveur. = I made them a favor. = I made a favor to them.
Mes devoirs, je les ai faits. = My homework, I did it. = I did to my homework.

If you are speaking of something receiving the action (COI, see Begueradj’s answer), then you will use leur. If the pronoun is the object of the action (COD, same answer), then you will use les.