"The only thing more troubling than the silence were the whispers that followed."
沈黙
{ちんもく}
より悩
{なや}
ましいのは継
{つ}
ぐささやきだけだった。
沈黙
より悩
ましいのは継
ぐささやきしかなかった。
Your grammar is good, period. It is the couple of word choices that would need to be improved -- 「悩ましい」 and 「継ぐ」, especially the latter. (I will be discussing yet another word choice later.)
「継ぐささやき」 simply would not be understood by Japanese-speakers unless they read your English original as well. The verb to be used instead would be 「続{つづ}く」 and it should be used in the form 「それに続く」 with 「それ」 referring to 「沈黙」.
Next, 「悩ましい」.
While I will not call it the wrong word per se, a more natural word choice might be 「煩{わずら}わしい」 at least for us native speakers.
The third word choice that I feel like mentioning is 「ささやき」. While I could not call it incorrect or anything, I would wonder why you would want to use a nice and generally positive word here when it is the thing that is bothering you. I would personally use 「ひそひそ話」, but it would depend on the larger context/situation in which this sentence appears.
Put it all together and one would have:
「沈黙より煩わしいのは、それに続くひそひそ話だけだった。」