They are very similar in meaning when you want them to mean complaining. In fact, you can usually use them interchangeably. "Moan" and "groan" or "moan and groan" makes one think of the sounds people make when complaining, "grumble" makes one think of complaining under one's breath. Hope this helps, Ken! :)

I forgot to mention: "moan," "groan," and "grumble" are all onomatopoetic words. That is, the sound of the word itself is actually an imitation of the sound it is describing. Mooooaaaan, moooooaaaan, mooooaaaan. GROAN! GAHROAN! GROAN! grumblegrumblegrumblemumblegrumblemuttermuttergrumblegrumble

They are very similar. Literally, they are descriptions of nonverbal noises people make to express emotion. To say someone is "moaning, groaning, and grumbling" is a disrespectful way to say you don't care about their complaints, it's just noise to you. A "moan" is a smooth, almost musical sound, "Oooooo" or "ohhhh" or "ahhhhhh." People moan when they are unhappy or in pain but they also can moan with pleasure. A "groan" is harsh and usually indicates pain or irritation. "Grumble" means the person is complaining, but they are sort of mumbling their words and you can't understand them. Or, they can mean the person is complaining but you've heard it before, you don't care, and you aren't listening. Moaning: "Ohhhh, nooooo.... not agaaaaaainnnnn." Groaning: "Arrrgh! AGGGGGG! I can't STAND it." Grumbling: "i keep telling them and telling them that i do not want coffee in a demitasse after the meal i want coffee in a full sized cup with the meal I want coffee with the meal is that such a big thing to ask for no it isn't its reasonable so why can't I get it but it's impossible you can explain it until you're blue in the face but can you get coffee with the meal no you can't they always bring it afterwards in a demitasse I'm telling you I don't know what the world is coming to it shouldn't be that hard to get coffee with the meal at a fancy restaurant but..."

You can moan with pleasure. Groan has a connotation of moaning with displeasure. And grumble is a combination of mumble and groan, so it is when someone is complaining under their breath, almost inaudibly.

For my part, moan and groan are synonyms. It’s an utterance expressing pain or disapproval. To grumble is to complain about somebody or something in a bad-tempered way.