Learn Croatian: Jedva čekam ljeto! - I can't wait for summer!

Jedva literally means "barely" or "hardly". The Croatian idiom for being excited about something translates to "I barely wait" (jedva čekam). If you were to use ne mogu čekati, it would mean you literally do not have the time to wait around, rather than showing excitement.

Why isn't there a preposition like "for" before ljeto?

In English, you wait "for" something, but the Croatian verb čekati (to wait) takes a direct object right after it. You do not need a preposition; you simply say "I wait summer".

What grammatical case is ljeto in this sentence?

It is in the accusative case because it is the direct object of the verb čekati. Because ljeto (summer) is a neuter noun, its accusative form looks exactly the same as its basic dictionary form (the nominative case).

How would I use this phrase to say I can't wait to do something?

You can just put an infinitive verb right after the phrase, such as jedva čekam ići (I can't wait to go), or you can use da plus a present tense verb, like jedva čekam da idem.