I received an email from someone who (like me) finds the Symbol Challenges on The Da Vinci Code Quest on Google to be pretty tough.  She asked me for some help solving these challenges, and I thought the rest of you might benefit from this as well. I've come up with a solution that generally works well as long as you're careful, analytical, and patient.  I'll share that approach as best I can in this article.

Here's the first puzzle Google threw at me when I decided to document my strategy for solving these:

The first thing you should notice is that the top row is almost completed in this example.  Google has given us a lot of help here.  Based on the fact that all 6 symbols have to be in this first two, I can deduce which two symbols are missing here and drop them in.  Since I can't tell from the other symbols already on the grid which of the two symbols go in which of the two spaces, I just take a guess and lay them down for now, knowing I can change them later:

Having completed the first row, and looking at all the other symbols already there, what can I deduce about which symbols need to go where in the rest of the puzzle? Quite a lot, actually:

1. I know that the "phi-looking" symbol can't go anywhere in the left-hand column because it's already there.
2. I know that the "fleur-de-lis" for the second row can't go in the second or third position. It can't go in the second because there's one already in that column. Even if there wasn't, I know it can't go there because there's already a fleur-de-lis in that colored area and there can't be two in it.  The fleur-del-lis can't go in the fourth column of the second row because there's already one in that column. That means it can only go in the first column or one of the last two.
3. I know that the Ankh symbol for the third row can't be in the third, fourth, or fifth columns because there's already one in those columns.  Similarly, I know that the Ankh symbol for the light gray area of the puzzle can't be in the second row, or the fourth row because those rows already have one.
4. I know that that the Blade (triangle) symbol can't go in the last column of the second row, nor can it go in the second column of the second row. That means it can only go in the fourth column or the first.

So, let's say I decide to tackle the "Fleur-de-lis" symbol first. The top row and second column already have one.  The third and fourth rows and the third and fourth columns already have one. So, to get one into the red area I will need to put it in one of the last two columns, in either the second, third, or fourth rows.  But since the third and fourth rows already have that symbol, I have to put it in the second row.

The first four rows now have a fleur-de-lis in them. That means in the first column the only place I can fit one is in the bottom left square, so I put it there.  That now means in the bottom row the only place I can put one is in the fifth column, because the first, second, third, fourth, and sixth now have one.  I end up with this arrangement:

Now, for no special reason, I decide to place the Greek Cross symbol.  The first column already has one.  None of the other columns or rows do. 

Since I need one Greek cross in the second column, AND I need one in the lighter gray area, there is only one place I can really put it. That's at the "L" part in the fifth row. So I do.  Now, the first and second columns are off limits and the fifth and sixth rows are off limits.

Looking at the dark gray area next, there's only one place I can put a Greek Cross. That's next to the fleur-de-lis that is already there, so I put it there.  Now the fourth, fifth, and sixth rows are off limits for the greek cross, as are the first, second, and fourth columns.

Looking at the third row and the brownish area of the grid, I see no greek crosses. I decide to place it between the blade and fleur-de-lis that are already there.  This means the first four columns and last four rows are now off limits for that symbol.

To get a greek cross in the next row, I decide to place it under the Ankh. (Note that this is a mistake because I already placed a greek cross in this column. I just didn't notice it.)

This now leaves only the last column and the first row as a place for the final greek cross, so I place it there.

There is where I am now:

Now I decide (for no special reason) to start placing the blade/triangle symbol.

I'll need one in the light-gray area of the first column.  The third row is off limits because there is a blade there already.  That leaves me with only the second and fourth rows. I decide to try the fourth row.

Now, I need a blade in the third column.  I only have two optons here, the fifth or sixth row. I decide to put it in the fifth row. 

That move leaves me with only one place in the dark gray area to place a blade, which is the sixth column at the last row.  I put it there.

At this point, the only free column is the fourth one, but I had a greek cross in there from last time around. This is when I saw my mistake, so I moved the cross to the right and put the blade down in its place.  I took a sanity check at this point to make sure I didn't have two of the same symbol in any row or column or colored area.  I didn't, as you can see:

Again, for no special reason, I decided to place the Ankh symbol next.  There are already Ankhs in the first, second, and fourth rows, as well as the third, fourth, and fifth columns.  Those areas area all now off-limits as are the matching colored areas in which they are contained.

In the first column, I decided to place the Ankh in the first available row, which is the third row.  It's also the only free row left where there isn't a symbol already and where an Ankh could possibly go.

In the second column, the Ankh could only go in the second column, so I put it there.

The third, fourth, and fifth columns already had Ankhs, so I didn't put any there.

The sixth column needed an Ankh, and the only place left it could go was the fifth row, so that's where I put it.

I was now here:

Now it was time to place the "phi-looking symbol" down, I decided.  There was one in the first column already.

The second column had none, and I needed one in the brown area, so I placed it in the second row. This left the second row and column and the first row and column off limits.

In the third column, I decided to place it in the last row. Now the first three columns, first two rows, and last row were off limits.

In the fourth column, There was only one space left, so I put it in the fifth row in that remaining space. 

In the fifth column, My only option for getting one in was the third row, so I used that.

In the last column, the only option was the fourth row, so it went there.

This left only the key symbol, and put me here:

I began placing key symbols in the remaining spaces and, sure enough, the puzzle was complete!

I then went on to my question and was done for the day.

I hope that by walking you through this puzzle step by step and explaining my thinking along the way, I've helped you sort out how to solve these puzzles.  This is the exact approach I use, and it generally fails only when I screw up and forget to take into account the rows, columns, and colors needing to have only one of each symbol in them.

Good luck in your questing!