I think you could probably read Destiny without any of the other TNG-Relaunch/Titan books. Reading the other books would help, so if you can get ahold of them I'd say wait. But if you can't or just really don't want to wait you could probably read the trilogy with very little difficulty.
It will greatly enhance your enjoyment of Destiny of you read, at a minimum, TNG's Before Dishonor and Greater Than The Sum (on a personal note, I can't in good conscience recommend Resistance, which I found a snoozefest, and which was recapped quite well by BD and GTTS); while Destiny does a fair job at recapping events of BD and GTTS in a few paragraphs reminiscent of (cue Majel Barrett's voiceover) "Last time, on Star Trek:The Next Generation...", to get the full impact of these events preceeding Destiny, read those two books.
And, no, Peter David and Christopher Bennett aren't paying me to shill their books...
If you get hooked back into TrekLit by Destiny, the events continue to unfold in Titan:Over A Torrent Sea and the stand-alone A Singular Destiny, as well as the Voyager book Full Circle (which covers times before/during/after Destiny). To that end, reading at least the first Titan book, Taking Wing, will familiarize you with the cast and crew of that ship, which participates heavily in the events of Destiny. Destiny takes place between Over A Torrent Sea, the fifth book of Titan, and Sword of Damocles, the fourth book, so a reading of SoD might be in order to bring you up to speed on Titan immediately before Destiny. I did not read SoD, but had read two other Titan books, so I was just fine.
So, to recap, the Official SicOne Recommendation, in order...
Titan:Taking Wing
TNG:Before Dishonor
TNG:Greater Than The Sum
Destiny trilogy
A Singular Destiny
Titan:Over A Torrent Sea
Voyager:Full Circle
Hope this helps