This is a great law review article by a judge currently on the US 9th Circuit. His thesis is well expressed by the opening paragraph:

Although we pretend otherwise, much of what we do in the law
is guesswork. For example, we like to boast that our
criminal justice system is heavily tilted in favor of
criminal defendants because we’d rather that ten guilty men
go free than an innocent man be convicted. There is reason
to doubt it, because very few criminal defendants actually
go free after trial. Does this mean that many guilty men are
never charged because the prosecution is daunted by its
heavy burden of proof? Or is it because jurors almost always
start with a strong presumption that someone wouldn’t be
charged with a crime unless the police and the prosecutor
were firmly convinced of his guilt? We tell ourselves and
the public that it’s the former and not the latter, but we
have no way of knowing. They say that any prosecutor worth
his salt can get a grand jury to indict a ham sandwich. It
may be that a decent prosecutor could get a petit jury to
convict a eunuch of rape.

He goes on to detail many of the specific issues, and even better to offer detailed suggestions on how significant improvements could be made. Many of those that have to do with juries are seemingly obvious:

1. Give jurors a written copy of the jury instructions.

2. Allow jurors to take notes during trial and provide
them with a full trial transcript.

3. Allow jurors to discuss the case while the trial is ongoing.

4. Allow jurors to ask questions during the trial.

5. Tell jurors up-front what’s at stake in the case.

6. Give jurors a say in sentencing.

In my limited experience, I think providing the jurors with a full transcript of the trail might make the most positive influence.

In the current system (at least where I am in California) we are allowed to take personal notes, but there is no reasonable way to match the notes of the individual jurors to the testimony that actually took place. A court stenographer makes a transcript of the trial while it goes on, but this transcript is considered to be their personal copyrighted property. If the jury wants to hear a portion of the testimony again (perhaps because the jury members disagree as to what was said), a written request must be made to the judge that describes the portion of testimony that the jury desires.

The following day (or whatever day the original court stenographer is available, which may be longer) the stenographer appears in front of the jury to read aloud from their transcript the portion that the judge feels is appropriate. The reading is done a single time from point A to point B without intonation and about as fast as the stenographer can speak. If they can keep up, the jury can take notes on this, but no typed copy is made available, ostensibly because this would violate the stenographer's copyright. Requests for portions to be read more than once are met with eye-rolling, initial partial compliance, and then eventual refusal due to scheduling constraints.

Can anyone believe this is the system which best enables a jury to reach the truth of the matter? Consciously or not, it's designed to have jurors decide based on their gut instincts instead of on the fine details of the case.