Muka muka (むかむか) is anger building up inside you.
Someone eats the last piece of cake you were saving. You don't shout, but something hot rises in your chest. That is むかむか.
What it means
むかむか is anger that keeps growing inside you: you are annoyed, it won't go away, and it is hard to put into words.
How to use it
Use it with します: むかむか します, むかむか しました. The slang verb むかつく (“that makes me so mad”) comes from the same word.
けさ、でんしゃで ひとが ぼくの あしを ふみました。
This morning on the train, someone stepped on my foot.
えっ、だいじょうぶですか。
Oh no, are you OK?
あやまりませんでした。 まだ むかむか します。
They didn't even say sorry. I'm still fuming.
Close cousins
いらいら is impatience: waiting, traffic, things that won't go right. むかむか is anger at something someone did. ぷんぷん is cute, pouting anger, the kind you are not really serious about. Fuji's pouting picture on this site is called ぷんぷん.
In manga and anime
An angry character gets a red cross-shaped vein on the forehead and ムカッ or ムカムカ beside it. People in anime say むかつく all the time.
Can you use it?
むかむか is the anger you feel inside, the kind that is hard to put into words. You tell a friend about it later; you don't say it to the person who made you angry. むかつく is rude: keep it for close friends, and never say it to a teacher or a boss.