How Do I Remove Diamonds Out Of A Pendant?

I have in the past had a diamond ring with a similar problem. Here's what I did: Since the prongs that hold the diamonds in place are so small I had to take a needle and go between the diamond and the prong then very gently at an angle pry until the prong was no longer touching the diamond. You will have to do this for each diamond until the prongs are no longer holding tightly on the diamonds then they should fall right out. Note: Please do this over a black or dark cloth as not to lose the diamonds. 

Why not try a reputable jeweler? I had a james allen diamond engagement ring and for some reason I didn't need the ring anymore. The ring was too valuable to give away so I had the diamonds taken out. This way, you will be assured that your diamonds will be in good hands.

If they are baguettes, they are very likely channel mounted
and none of the above suggestions will work since they are
set in solid metal. I am surpised someone told you they were
more valuable out of the setting. If it were me I would (do it
myself - seriously not recommended if you don't know what
you are doing) or take it, or send it, to a REPUTABLE gold
refiner who offers "stone retreival services". For usually less
than $50 they will literally dissolve the gold metal from around
the stones resulting in ZERO damage to the stones and pay
you for most of the gold content of the ring as well.

Apparently you or the person who bought it for you is freakin rich to buy diamonds like those!!!!