Well, given the complexity and flexibility of human languages (and underfunding of the Grammar Police Force), I think that looking for a quick & easy method is usually setting yourself up for disappointment.
Quick & easy method: Adverbs modify (give you (additional) information about) verbs, adjectives, or other adverbs. An adverb clause is a subordinate clause that functions as an adverb. So when you locate a subordinate clause, ask yourself if it modifies a verb, adjective, or adverb. If it does, you have yourself an adverb clause. Congratulations!

Well, the instructions are quick and easy; I doubt that following them is either.
Notice that the definition (do you have a different one?) defines adverbs by
what they do / what they mean (function), not by
how they look / where they are located (form). So if you're looking for a way to identify an adverb by it's form, sorry, I don't know if that's possible. You might just be stuck looking at function or taking your chances with tricks, tips, rules of thumb, etc.
--BTW, I think this is a good general lesson. Instead of thinking, for instance, "this word is an adverb", it's better to think "this word is
functioning as an adverb
in this particular sentence" or even better "this structure (word, phrase, clause, etc.) is functioning as a member of this category (adverb, noun, verb phrase, etc.) in this particular structure".--
Here are the adverb sections from two of the best grammar sites that I've found: http://grammar.ccc.commnet.edu/grammar/adverbs.htm and
http://www.ucl.ac.uk/internet-grammar/adverbs/adverbs.htm. They give some tips -- they aren't necessarily foolproof though.

P.S. Why didn't you put this in the Social Sciences forum? Or History and Humanities even? Just wondering.
P.P.S. Subordinate clauses aren't always preceded by a subordinating conjunction, a.k.a. complementizer. I think this sentence demonstrates that fact. Sorry, Grammar Police. I think
that that sentence demonstrated that fact.
P.P.P.S And that last P.S. demonstrated the problem with thinking things like "
that is a complementizer", since
that functioned as both a complementizer ([I think]
that [that sentence...]) and a determiner ([
that sentence], [
that fact]).