Difficult adverbs in French: cases where -ly is used in English, but -ment is not commonly used in French

accordingly: When it means consequently, a possible translation is therefore par conséquent. When it means in accordance with what is expected/predicted/demanded, possible translations could include comme prévu or en conséquence. For example, to act accordingly is usually translated as agir en conséquence.
consequently: Usual translations are par conséquent, en conséquence de quoi (literally "as a consequence of which"), although the word conséquemment does also exist in French. Another, more formal, phrase is par voie de conséquence.
curiously: At the beginning of a sentence, possible translations include Chose curieuse..., Le curieux, c'est que.... Compare interestingly.
effectively: When used to mean "in reality", "when it comes down to it", then possible translations include en réalité, en fin de compte or, in informal usage, grosso modo.
interestingly: When used at the beginning of a sentence to mean "The interesting thing is...", possible translations include Chose intéressante..., Ce qui est intéressant, c'est que..., L'intéressant, c'est que....
strangely: At the beginning of a sentence, this can be translated in a similar way to curiously. Another possibility is Chose étrange....
worryingly: At the beginning of a sentence, a possible translation is L'inquiétant, c'est que... or Ce qui est inquiétant, c'est que....